The only true "glitch" I can think of that Melee has is wavedashing, although that's a pretty big stretch...I'd call it more of an exploit of the physics engine in the game.
The majority of the other advanced techniques were obviously intended to be in the game (judging by the fact that they were all in the previous Smash game, SSB for the N64); teching, L-canceling, dash-dancing, shield-canceling, etc., were all intentionally put in by the developers, in my opinion. At the very least, they were AWARE of them since they were in SSB64 as well.
Other than that, Melee also expanded a bit on edgeguarding (giving characters a "jump" option while recovering from a ledge and also extending the amount of time that a ledge is "ungrabbable" by a recovering character after being edgehogged by an opponent) and added dodging (both normal dodging and directional air dodging) to the gameplay.
Thus, as a whole, Melee had more options available to the players. As was mentioned before, the "advanced techniques" are actually not very advanced at all - they just require you to KNOW what they are and practice them like you practice the "normal" techniques of the game. Since most people don't know about them when they first start playing, therefore, they're called "advanced."
As for things like waveshining, drillshining, chaingrabbing, and other "cheap" things, you can't really call those "glitches" either. Again, like wavedashing, they're more like exploits - in this case they're just repetitions of certain move combinations that turn out to be extremely effective. However, they're not exactly "game-breaking" and there's always a way to avoid or counter them as well. If you actually take the time to look at what each one consists of, it's easy to see that they're not really "glitches", either: Waveshining consists of shining your opponent with Fox, wavedashing close to them, and shining again (rinse and repeat). Drillshining is similar: use Falco's down-air on an opponent, L-cancel, shine, and then either hit them in the air or knock them back down and repeat the process. Chaingrabbing is just "grabspamming" with a bit of prediction involved as to which way your opponent will DI after being thrown.
Many people will look at these things and say "Oh, man - that's really 'advanced.' " Well, no - it's not all that advanced if you KNOW it exists and practice it consistently along with everything ELSE that you practice. Certainly some techniques are more "technically" difficult than others, but on a whole I'd say that the majority of advanced techniques are not actually that hard to pull off - they just require practice and consistent incorporation into one's gameplay, which is something that Brawl does not offer.
So at the end of the day, Melee really doesn't have any glitches in competitive play. I'm sure you've heard of the Link-boomerang glitch or the Black Hole glitch; THOSE are definitely glitches, but you don't have to worry about them showing up in competitive gameplay. Melee IS imbalanced, as has been brought up, but that imbalance does not stop players from mastering low- or bottom-tier characters and thrashing higher-tier characters just the same...it's just a whole lot more difficult for the low/bottom-tiers to win in tournaments. Who cares; some people like the added challenge of playing "against the odds" (I'm a bit like that) and the game is relatively well-balanced between the 8-10 characters that are normally used in tournament play. It could be better, I know, but it's not, and if you're looking for competitive Smash play, you'd best be thankful that you've still got Melee and that Brawl wasn't the ONLY Smash game ever made.