Worst mechanic?
Wheeeeeeeeeeeee, that's hard. I've seen so many, many, many really bad ones.
OK, so...
1. Anything in a game that's designed to prevent the player's power from growing via time costs, in general.
This is almost always lazy design.
At worst, it's obvious padding to games with thin, weak designs; at it's best, it's faired into the power-curve the game is throwing at the player (in terms of increasingly-difficult / buffed things) and it may as well not be there at all; instead the game could just give the players <meaningful choices> when the game gets harder.
There are literally no good justifications, from a purely game-design POV, for "experience" systems; they're just trashy ways to make players stay at power plateaus, regardless of how well / badly they actually play the game.
Very few games get this right (in the sense of making it feel like it's not terrifically important, blocking, or just wasting your time). Most end up on the grindy end of the scale, where if I can't mod it out, I quit playing (or if it's really awful, write a scathing review). Game designers who don't respect our limited time on this planet because they don't know what else to do to create a difficulty curve are pretty common, unfortunately.
I think that's my no. 1 sin, ever.
2. Games that pretend that RNG suffices in place of actual complexity.
When you throw down a bunch of RNG, the results are inevitable noisy trash, not beautiful complexity.
This is in part due to how computerized RNG generally operates; sometimes, for a few kinds of game designs (card games, for example) the developers actually build complex RNG systems that have better noise systems than just Math.random() but this is not common (because it can be computationally expensive).
In general, games that rely on RNG to create a lot of the "fun" are bland, forgettable and lazily designed in general. Cool mechanics are, like, actual work.
3. Games that are fairly balanced except for a few obvious Right Answers.
This happens a lot, especially in aRPG systems; as they grow in complexity and the core systems are mature and content keeps getting out of the pipeline, designers tend to lose track of the fundamentals, because it's soooooo boring to have to do more testing of <insert boring scenario> to make sure that the Dagger of Luckiness +3 isn't, say, a replacement for every other Dagger in the design. When I encounter these things in a game that's otherwise competent, I immediately want to mod it out of existence.
Games where that can't be done (looking at you, Far Cry series) make me grumpy, because often the fixes to get it into the "close enough" range are fairly small, like narrowing the ranges of enemy health or making a weak thing a bit more competitive and so forth. These kinds of things happen more often with AAA than they used to; it used to be that AAA meant a lot more playtesting before release and a lot of competent eyes on the product, but nowadays they rarely seem to take the time / spend the money; they'd rather blow another million dollars on kewl Content to distract us, rather than <yawn> reviewing the existing stuff critically in the context of actual playtesting.
4. Games where a core mechanic was very poorly tested / polished before release.
An excellent example of this would be the "knife" mechanic in Teleglitch. Teleglitch, by design, is about not having ammunition for your firearms, so you're supposed to stab things to death.
The designers were told (by a third-party game designer from a college specializing in same) that they had a serious problem, in that it wasn't much Fun for players to not have a way to defend themselves when they were in scarcity situations.
Their solution? An infinite-ammo weapon the player could wield; short-ranged and weak, but it was better than nothing at all (yup, the knife was added late, as a fix for something any player of Doom could've told them was a problem with their concept). However... the knife, while technically functional, is very poorly thought-out; it's a raytest that occurs on one frame that only covers a small distance, in a straight line, for one gameframe.
Long story short; it's incredibly hard to actually use, unless you have godly reaction times and know the hitboxes really well.
This mechanic could've worked with minor tweaking (make it last more than one gameframe, make it affect an arc, make it a little longer) but it wasn't. So, the game's difficulty, already rather overwhelming for most players, in a niche market to boot, was made frustrating; people expected a mechanic like that to work in a fairly forgiving way, if poorly, and instead, it just leads to one-mistake-and-dead gameplay.
Sometimes, things hang on one really unpolished (but core) mechanic. I've seen games where jumping was supposed to be important, where the jumps aren't quite right (or are suddenly darn-near impossible), games where they nerfed their BFG into nothing more than a pretty light show, rather than providing a tool with limited use cases (always interesting, if done well) games where you're supposed to jump in and out of vehicles, but that process is clumsy or ill-behaved (and games where they did it brilliantly, like Saints Row 3).
I have no idea why it went wrong, in this case (there isn't a lot of post-mortem on that game), but I'm guessing that it was added at the last second, under duress, by the brothers who co-wrote it, because they didn't see any problem and felt annoyed that they had to put it in. But it's always one of those exemplars of "how to hose an otherwise-good idea" in my mind.