How does shield bashing work?
I've looked in the DMG and the PHB, but have only found ancillary uses of the shield. The Shield Master feat (PHB, p.170) gives you neat perks, but doesn't act as a weapon (beyond the bonus shove action).
Does shield proficiency automatically grant you shield bashing?
What would reasonable damage be?
And do the old rules apply for losing the defense bonus when using it as a weapon?
Does it count as a light weapon? Can I use it as a two-weapon bonus action?
A general preface: each edition of D&D is its own game, and rules should be evaluated as such. Just because things happened in old rule sets does not mean that it will in 5e. Though it can be helpful to look at old editions for inspiration, you should be careful of the rules environs of the edition you are trying to modify when making house rules.
First, Shield Bash is already a named mechanic in 5E: it's the additional power associated with Shield Master that allows you to attempt to push someone as a bonus action when you make an attack. Also, you can do this same thing as a Full Action (or a piece of an attack action) using the contest rules and the "Shove" improvised action (Players Basic 74). That said, that's not what you're asking about here. What you want to know is how to do damage with your shield.
If you want to hit someone with your shield, it is treated as an improvised weapon attack.
The question then becomes does it resemble any existing weapon we have game statistics for? as per the "Improvised Weapons" section on pg 47 of Player's Basic. Looking at the list, I don't see an obvious resemblance (though feel free to make your own observations/additions). Thus it acts as a 1d4 weapon and proficiency is only granted if you have proficiency in improvised weapons.
There is also no provision for removing the defense bonus. In fact Shield Master improves the shield with attacks without removing the defense bonus so there seems no need to import that rule from a prior edition. (Additionally, it seems to be punitive beyond the spirit of 5e's rules.)
So, ultimately, shield bashing is a flavorful power that is probably not a good mechanical choice. If you want to do it, talk to your DM, but without house rules, it's not a great use of an action. If you want the mechanics, and don't mind not doing damage on the push, Shield Master's push is probably the best you're going to do.
But would the fact that a shield is a martial device make it more than an improvised weapon? I agree with the flavorful power, that's a good point. But I would argue that it should be about 1d6+str and not be considered an improvised weapon. Or maybe I've just been playing Skyrim and Dragon Age too long.
@GMScherz There's nothing in the game that allows for that at this time. What it *should* be is between you and your GM. The mechanics on improvised weapons allow those that closely resemble weapons to be used as such, thus you might talk your DM into letting you use your shield as an ax or something, but I'm not seeing a *close enough* comp to include it in an answer.
From a HEMA perspective, shields *are* weapons; not just passive things used only to absorb ill-placed blows. 1d6B damage would be appropriate considering actual shield use. D&D, however, has a history of totally ignoring anything to do with the real world, no matter how cool it actually is! Therefore, this answer stands.
@PipperChip The HEMA perspective is flawed if "actual shield use" leads you to conclude that a shield does as much damage as a shortsword. If that were the case, shortswords would look more like these deadly shields you speak of. Shields were rarely used to try and cause damage to an armoured opponent, because they were not designed for that, making it a very high risk, low reward strategy.
@GreyWulf This is a fantasy game. In reality being stabbed with a shortsword is just as deadly as being stabbed by a longsword or a sufficiently pointy steak knife. Game balance does not necessarily equate to real world application.
@GreyWulf Even been hit by one? ;) DnD also has unarmored people that can take a full-on halberd hit and still walk away in one piece. Hit points are actually very bad at tracking deadliness, because it's as Greedy Radish said: it's not so much the weapon as the application.
@greywolf If you want to get technical about armored opponents, most swords did very little vs heavy plate, and what they did was not slashing damage.
Minor correction: shoving is an attack, and can be made as a part of the Attack action (PHB p. 195, Melee Attacks)