Weapon Mastery was hyped a bit in the run-up to the 5th playtest packet for One D&D, but honestly, it's one of those things that gets cooler the more I think about it.
We have yet to see how Monks will work - they're the last remaining class whose first revisions we have yet to see - but we can at least comment on these two.
While Fighters can learn to swap out these masteries, we'll be building around the default ones to accommodate Barbarians and, later, Monks.
Let's go through the broad archetypes of weapon loadouts for these two classes. In melee, Barbarians and Fighters have similar options, though I think Barbarians will like to dual-wield more given their Rage bonus and the Fighter's extra attacks, the latter of which dilute the additional damage of a single other strike.
Strength-based builds come in what I'll call four varieties, which Fighters and Barbarians both might use. These are:
Dual Wield
Sword-and-Board
Great Weapon
Long-Range Polearms
Barbarians are locked into using Strength builds due to Rage, but Fighters can also go for Dexterity builds. These come in the following forms:
Dextrous Duelist
Dextrous Dual Wield
Marksman
Ok, with those as the archetypes we're going to look through, let's pick what masteries will be most appealing.
Dual Wield:
A dual-wield build focuses on light weapons. There are two weapon masteries that the Light property makes possible, and another that can go on any weapon. These three properties are Slow, Nick, and Vex.
Here's what's really interesting: I don't think you need to have Nick on both weapons in order to gain its benefit. Given that this property only affects your off-hand attack, you can probably only worry about having a Nick-capable weapon for your off-hand. The only martial weapon with Nick is the Scimitar, which has the highest damage die at 1d6.
For our main-hand weapon, our two best options are the Shortsword and the Handaxe. Oddly, the Handaxe, despite being a simple weapon, is almost strictly superior to the martial Shortsword, given that it has the thrown property while otherwise having the same statistics.
So, I think Handaxe and Scimitar is probably your best bet here, unless you hate slashing damage.
Sword-and-Board:
Here, we want a one-handed melee weapon, but that's our only real requirement. This rules out masteries that require a two-handed weapon, and effectively any heavy weapons with one very interesting exception.
Actually, let's cover that exception now: you can use a Lance in one hand if you're also mounted. This will do more damage than other one-handed melee weapons. The Lance has the Topple mastery, which we can also get on other one-handed weapons, so we'll just fold it in as a special case.
Ok, so now we can talk about other properties we can get. While some could be swapped around by a Fighter, we're only going to look at the standard masteries of weapons we can use. These include Slow, Nick, Vex, Sap, Flex, and Topple. (Interestingly, while Push can be put on versatile weapons, none actually have it inherently.)
So, then, which of these are the most appealing?
If we're using a shield, we might be eager to focus on our protective, defensive capabilities, which means we might like something like Sap. For that, we'd take a Mace, Flail, or Morningstar. Of the three, the Mace is strictly worse than the Flail, as it does less damage while being otherwise more or less the same.
If our goal is not strictly avoiding damage, we might instead want more battlefield control. In these cases, we might go for Topple or Slow. Topple, on a success, will likely inflict a more grievous condition, but can be saved against. Our Topple options are the Battleaxe and the Trident, while our Slow options are the Club, Javelin, and Whip. Notably, the Slow melee weapons are all lower in damage. The Whip does also give us reach, but it and the Club are both only d4 weapons. So, for this, I think the natural choice would be to go for the Battleaxe or Trident, and the latter is generally slightly better, due to the fact that it can also be thrown.
Note here that Lances will actually come with a lot of benefits, and become the best one-handed option, but only if you're fighting from a mount. In this case, the Lance becomes a reasonable alternative to the Trident, and pulls ahead of the Battleaxe.
Now, if we want to maximize our damage, perhaps to balance out the sacrifice from using our off-hand for a shield, Vex or Flex are our two best options. Flex is found on the Quarterstaff, Longsword, War Pick, and Warhammer. The Quarterstaff has lower damage than the others, so we'll generally want to pick from one of the three other options, which conveniently also give us one option of each of the three main weapon damage types, so choose the one you like best. For Vex, our options are the Handaxe, Shortsword, and the Rapier. The latter deals more damage, but the former has the thrown property, so this depends very much on whether you are willing to do slightly less damage for the ability to throw your weapon. The Shortsword winds up being the loser here, as aside from its different damage type, the Handaxe, despite being a simple weapon, is actually strictly better, with the same properties and then thrown in addition.
Nick is irrelevant to us given that it's only for dual-wielding.
Great Weapon:
The idea of this fighting loadout is to use the highest-damage melee weapons in the game.
Here, we want to maximize the damage we deal with the heaviest weapons. The highest-damage-potential melee weapons are still the Greataxe, Greatsword, and Maul.
The Greataxe falls slightly behind due to the fact that 2d6 does slightly higher average damage than 1d12 (basically because its lowest roll is a 2, rather than a 1) but this also gives you the Cleave mastery. This bonus attack benefits someone who has lots of bonuses to their damage outside of their Strength, and is diluted, like dual-wielding, by a class that already does more attacks. So, I'd say Barbarians are more likely to favor Greataxes.
Meanwhile, Greatswords have the Graze property, which allows you to deal your Strength modifier (interestingly, the very thing that Cleave doesn't) to the target even on a miss. This will be good against targets with high ACs, like bosses.
And then, Mauls have Topple.
This one's kind of a situational thing. Cleave is amazing when it can actually do something. Graze is a wonderful way to feel like you're always contributing something. I don't think there's a right answer here. Mauls will work equally for both, but I'd say Cleave slightly favors Barbarians while Graze slightly favors Fighters (by level 18, you could miss four times in a row and still deal 20 damage to a target).
Long-Range Polearms:
If we're looking to make use of Reach weapons, we have actually a fair number to choose from. Glaives, Halberds, Lances, Pikes, and Whips.
The masteries, then, that we get to pick from are Graze, Cleave, Topple, Push, and Slow.
The Glaive/Halberd choice here directly parallels the Greatsword/Greataxe discussion, but I will say that if you go with Polearm Master, Graze (from the Glaive) will shine a little more on your bonus action attack given that the damage die is so low anyway - you were mainly there for the Strength bonus anyway.
For battlefield control, the other three masteries are each on one weapon. Before we continue, though, we need to remember that Slow is just on the Whip, which is probably not what we're going for with this build, and probably instead using it more in a sword-and-board situation. The Lance gives us Topple and the Pike gives us Push, so whichever you think will be more useful to you, you can pick it. I will say, though, that with our longer reach, Push might wind up being more useful, as we can really hold off a foe this way while remaining safe. Indeed, Push can be very good with the "brace" opportunity attacks you get with Polearm Master, which you probably are getting if you want to play this way.
Dextrous Duelist:
We're now looking at melee-dexterity-fighters. The Dex Duelist is going to be fighting with a finesse weapon, but is not dual-wielding, and likely instead has a shield.
A lot of the same considerations from our Sword-and-Board discussion is relevant here, but now we need to only fight with finesse weapons.
Once again, Nick weapons are irrelevant here. But among one-handed finesse weapons, we can also get Vex and Slow.
If we want Vex, our options are Rapiers and Shortswords. The only thing Shortswords have over Rapiers is that they are light weapons, but if we're using a shield, we don't need that, so Rapiers become the clear option if we want Vex.
The only alternative is getting Slow with the Whip. This also has reach, and is actually a very solid choice if you want to use the range and slow effect for battlefield control, but your damage will definitely suffer, dealing 2 less damage on average per hit
Dextrous Dual-Wield:
Once again, we want to use an off-hand weapon with Nick, but this time we need it to be a Finesse weapon, which cuts Sickles and Light Hammers out of the conversation. What's left here are Daggers and Scimitars. Again, daggers can be thrown, so they could be a viable (and classic) option for a "main gauche" weapon, and only deal an average of 1 damage less than Scimitars, so it's actually a pretty viable option.
For our main hand, though, the other light finesse weapons all have Vex. These are... let me check my notes here... just Shortswords. So... yeah, you're going to use a Shortsword.
As a note, technically you could swap these around - I think you only need one or the other light weapon to have Nick to benefit from it, so you could have a Scimitar in the main hand and a Shortsword in the off-hand, but given that Vex can activate multiple times a turn, I think it's best to have that on your main hand weapon.
Marksman:
So, now we come to ranged weapons. While there are a lot of options for ranged weapons, there are only a few masteries that are associated with them, and I think your choices are going to be more about the other properties.
Ranged weapons only have Vex, Slow, and Push. All of these are relevant to ranged characters - Vex will, of course, translate to hitting more often, but Slow and Push will both potentially make it easier to keep enemies at range and preserve your safe position.
However, I'd say that Push is more powerful here, given that, as a ranged character, you don't have to chase the target to keep pushing them, and if you're trying to set enemies back, you can get more with multiple attacks.
The only weapon with Push on it natively is the Heavy Crossbow. This is a decent option, but remember that you'll need to invest a feat in it to make sure you can shoot this more than one time per turn due to the Loading property. Crossbow Expert will open up a lot here.
If you don't want to commit to the Heavy Crossbow but still want that battlefield control from Slow, you have a lot more options: Light Crossbows, Slings, Longbows, and Muskets. Among these, the Musket does the most damage with 1d12 - the highest of all ranged weapons - but it also has the Loading property. If you can pick up the Gunner feat, which has not yet been confirmed to appear in the new PHB (though theoretically you should be able to pick it up from Tasha's given backwards compatibility,) this becomes pretty great, though.
Bear in mind, though, that firearms actually have fairly short ranges - the regular range on a Musket is only 40 feet, so you might also need to invest in Sharpshooter to fire normally at its long range, which is 120 feet.
The Light Crossbow does less damage than a Musket, and would need the Crossbow Expert feat to ignore its loading property. The Sling does less damage and has an even shorter range, but doesn't have Loading. But these are simple weapons, and naturally do lower damage. Among the martial weapons that have Slow, the only other option is the Longbow. The Longbow deals less damage than the musket, but its range is far, far longer - in fact, its short range is longer than the Musket's long range. There's also no need to invest any feats to get it to work with Extra Attack.
Now, if we instead want to get Vex, we have other options. These include the Dart, Shortbow, Blowgun, Hand Crossbow, and Pistol.
For pure damage potential, the Pistol easily wins with 1d10 damage, but its range is also very short - only 30/90, and with the loading property, you need a feat.
But:
The Blowgun somehow manages to be worse in nearly every respect, with a normal range of only 25 feet and a damage die of... a d1. That is to say, it just does 1 plus your Dexterity modifier. Technically, you can hit farther targets than the Pistol, as it has a long range of 100, but you're looking at an average of 4.5 lower damage with each hit, and no feats that let you ignore loading here.
I have no idea why the Blowgun is a Martial weapon. It's barely a weapon at all.
That aside, let's look at Darts. Darts are thrown-only, letting you get a dagger-like effect but with Vex instead of Nick. This is a weird edge case that I honestly think could be ok for Rogues that take the Weapon Master feat, but probably doesn't make sense for most others. And you can have a lot less fuss and a better range with a Shortbow.
The Shortbow is probably the best no-fuss ranged vex weapon. But then we have the Hand Crossbow. This is the only ranged weapon with the light property, and while I don't know if there are still issues with whether you can dual-wield these even with Crossbow Expert (or if even ignoring the loading property still requires you to be able to draw another bolt) and the new version of the feat doesn't just let you fire the same hand crossbow, this is at the very least a way to have a melee/ranged mix.
So... um... what is our conclusion?
Basically, unless you have to cut out Heavy weapons, your best choices as a Ranged dex build are probably going to be the Longbow, the Musket, the Pistol, or the Heavy Crossbow, though only the firearms if you have access to the Gunner feat or some other way to ignore the loading property. The Hand Crossbow could also be a viable option for a dual-wielding ranged build, though we'll need to see if this receives more clarification (in fact, I think it's not the Loading property that creates problems for it, but just the Ammunition property). The Musket, Pistol, and Heavy Crossbow will all require feats to fully make use of them, so if you're averse to delaying maxing out your Dexterity, the Longbow will serve you well. But these other options can be very good as well (while Sharpshooter is also probably a must for all of these, you'll probably want to grab it as soon as possible if you're using Firearms so that you can get a decent range).
Now, as a note, the Fighter feat progression now lets you pick up a feat at level 5, right after getting one at level 4. As such, you could wait on Crossbow Expert or Gunner until level 5 and grab Sharpshooter at level 4 - and these two should combine to give you a total of +2 to your Dexterity, letting you nearly keep pace with other characters. There is one redundancy, as Crossbow Expert, Gunner, and the new Sharpshooter all prevent you from suffering disadvantage for shooting in melee, but each gives a lot of other important stuff.
There is no doubt that the weapon mastery system has introduced a lot more nuance to picking your weapons. I'll be curious to see how mandatory the Weapon Master feat will be for non-Warrior martial classes. Vex seems like it would be insanely important for Rogues to get their hands on, as it would seriously enable Sneak Attack constantly. Paladins and Rangers are also probably going to want to invest in these if they can.
Now, we don't know how this system will interact with, for example, a Monk's unarmed strikes. If they don't, the Monk will probably want to still mostly use weapons for their actual attack action. Bladesinger Wizards, melee Bards, and Blade Warlocks will also likely want to take a look at this.
Presumably, Monks will still only get proficiency in Simple weapons, so we'll need to revisit these a bit to look at their options. Nick could actually be really useful assuming that Martial Arts still uses a bonus action for their unarmed strike. But we just don't know enough to comment on that just yet.
I will say, though, that imagining playing with these on a Fighter or Barbarian has got me excited. You can look to the previous post where I imagined how I'd do things on my level 12 Eldritch Knight Fighter, and it genuinely sounds insanely awesome.