The following types have multi-hit moves: Normal, Fighting, Ground, Grass, Ice, Bug, Rock, Dragon, Steel, Water, Flying, Ice
Full List of Multi Hit Moves as of Gen 9 (Note: Word on left = move. Word on right = type)
Variable Multi Hit Moves:
Arm Thrust (Fighting)
Barrage (Normal)
Bone Rush (Ground)
Bullet Seed (Grass)
Comet Punch (Normal)
Double Slap (Normal)
Fury Attack (Normal)
Fury Swipes (Normal)
Icicle Spear (Ice)
Pin Missile (Bug)
Rock Blast (Rock)
Scale Shot (Dragon)
Spike Cannon (Normal)
Tail Slap (Normal)
Water Shuriken (Water)
Fixed (Set) Multi Hit Moves
Bonemerang (Ground)
Double Hit (Normal)
Double Iron Bash (Steel)
Double Kick (Fighting)
Dragon Darts (Dragon)
Dual Chop (Dragon)
Dual Wingbeat (Flying)
Gear Grind (Steel)
Surging Strikes (Water)
Tachyon Cutter (Steel)
Triple Dive (Water)
Twineedle (Bug)
Moves affected by the ability Parental Bond
Accuracy Dependent Multi Hit Moves
Triple Axel (Ice)
Triple Kick (Fighting)
Population Bomb (Normal)
EDIT:
Beat Up (Dark) number of hits depend on how many Pokémon are in your party.
Source