Three new fighting styles

[Sindeloke]

All of these are available to fighters, paladins, and rangers.

Cavalier
While mounted, you may use your bonus action to direct your mount to make a single attack.
Additionally, once per long rest, when you mount a creature or construct with fewer hit points than 6 x your level, you may grant it enough temporary hit points to reach that number.

Spoiler: Mounties

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Nothing complicated here. I don't find the Mounted Combat feat does enough for survivability for horses etc at mid-to-high levels, so I threw in a hit point buff. Rangers won't need it, but they get more damage out of the bonus attack so it's still a strong style for them.

Careful Aim
If you move five feet or less on your turn, you may use your action to make a single attack with a ranged or thrown weapon. If you roll a 10 or less on this attack roll, roll again and use the new result.
If you have the Extra Attack feature, your attack deals 1x extra damage for each additional attack you could ordinarily make (so double damage with one extra attack, triple damage with two, and so forth).

Spoiler: Sniping

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The math on this works out to +2.5 to hit, 7.5% crit chance (as opposed to the +2 to hit, 5% crit chance of Archery). I'm not sure that extra +0.5 is quite worth giving up an entire turn of movement, but given how potent a crit becomes, I'd rather err on the side of caution. Due to the pseudo-advantage interacting with Superior Critical, this becomes really devastating on Champions, but you still do more damage as a ranger with normal Archery and Swift Quiver.

Unarmed Combat
The damage die of your natural weapons increases by 1 step, or by 2 steps if your character level is 11 or greater. When you aren't using a shield or manufactured weapon, you may take the Grapple or Shove action as a bonus action on your turn.

Spoiler: Punching things proficiently

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"Unarmed strike" is just a normal natural weapon in our games, and has been for several editions. It simplifies things. Humanoids aren't automatically proficient with their natural weapons (not even humanoids like gnolls who have a bite attack or whatever), but the proficiency is available to anyone who can use martial weapons, and also rogues. For us it deals 1d3 for a Medium humanoid rather than just 1, but either way it goes to 1d4 with this style and then 1d6 at 11.

And yes, it steps on the monk's punchy toes really badly, but that's totally fine with me. I consider the monk's proper niche to be "incredibly mobile skirmisher who laughs in the face of hostile magic," which it totally nailed this time around. Hogging unarmed damage and battlefield control too is unreasonable and silly.

[DragonLordIT]

I really like the careful aim thing, although I wouldn't put it as a style; I would think you could use it as a rule or special kind of attack action.

[Sindeloke]

I actually considered that! But I'm not really comfortable with letting a player freely stack +2.5 on top of the already pretty significant +2 from Archery without a little investment, which would mean taking the re-roll off the ability if it were just a free universal thing. And the only thing I can think of that makes sense as a bonus for "take your time to aim properly" besides +to hit is +crit. And I am nowhere near good enough at statistics to figure out what kind of crit bonus would be balanced, especially if you allow rogues access.