These are all the non Steel/Ground/Rock Pokemon that can learn Stealth Rock, you can decide which are viable for competitive play.
Pineco
Clefairy
Clefable
Jigglypuff
Wigglytuff
Chansey
Pinsir
Mew
Dunsparce
Miltank
Blissey
Celebi
Kecleon
Deoxys (all forms)
Turtwig
Grotle
Chimchar
Monferno
Infernape
Piplup
Prinplup
Bidoof
Bibarel
Uxie
Mesprit
Azelf
Arceus
Druddigon
Source