They are pretty lackluster unless they're Whitney's Miltank or Meowth of Team Rocket.
Or well, Snorlax and Chansey/Blissey seem to pop up in competitive play every now and then.
Oh, and Smeargle with Dark Void also used to be kind of a big deal until it got nerfed.
I don't play competitively tho so I really wouldn't know.
Guess part of their problem is that they're usually amongst the first creatures you encounter and possibly also catch in the game (Ratata/Pidgey, Zigzagoon, Bidoof/Starly, Patrat/Lillipup/Pidove, and so on), and Pokémon you get early on usually don't end up becoming particularly strong even after evolving, which makes all of them appear weak.
Sure, exceptions exist since I recall having and using a Stoutland in B/W with pretty good results, and Pidgeot can be kinda beastly too (but that's also not pure Normal type. It's also just so beautiful and muhjastic in its simplicity that I just can't help but love it and so I'm probably biased), but on a whole, since they have no particular strengths or resistances (aside from Ghost, which also resists Normal), it usually seems like a better idea to just use something else instead, or something that's not pure Normal-type.