The most common is "Any %," also the most basic, it means finish the game, usually anything goes, glitch away.
You can also have no glitches category, 100% complete, no warps, no powerups, all <game item>s, etc. Sky's the limit. The community creates the categories.
There is, however, OoT Randomizer[1] which is really fun to play and watch. You can get end stage items at the beginning depending on your seed.
Over the years the Super Mario Bros community in particular has become more accepting of glitches, as otherwise both TAS and human runs would have hit the limit of how fast is possible. By allowing glitches there is now always a possibility that the runs can be further improved.
Indeed, which is why I said "usually."
I usually see "Any %" and "Any % no glitches," but I just dabble in watching speedruns. If I see "Any %" with no other qualifiers, I'm going to assume glitching is allowed.
I think it can be a bit arbitrary at times when a technique is considered "glitching" or not. The community decides. Which is fine.
I saw a speedrun of Super Ghouls and Ghosts that had a clever limitation - "pick up all weapons that show up." If you're familiar with the game, if you pick up a weapon you lose your previous weapon and some weapons are way worse than others.
SethBling turning Super Mario World into Super Flappy Birds on a console comes to mind. https://www.youtube.com/watch?v=hB6eY73sLV0
But the latter is still not quite a "glitchless" run, based on the average person's definition of "glitch". For example, I doubt it was intended behavior for Kraid to die after 4 missiles [2] based on the player shooting him during a specific animation. I never played SM much beyond beating it as a kid, but I enjoy its speedruns because besides the variety of techniques and clutch moments required, there's at least one boss who behaves randomly, and requires the player to improvise. Also, in the course of new records being set, there have been a variety of strategies (e.g. different decisions on which path to take) employed, and there still seems to be room to discover more optimal strategies.
[0] https://wiki.supermetroid.run/Any%25_Glitched
Phantoon, bane of speedrunners.