During the scene where Alex pushes the guard who gets slashed by the trains' wheels, the guard falls backwards, so he's suppose to be lying on his back, but on the shot where Alex is standing next to him, the guard seems to be as he fell forward, and lies on his stomach.
When Alex goes to Rebel Skates he gets a board with white wheels. Later after the scene where Alex and Jennifer discusses to buy condoms, the board Alex carries is a different board with green wheels. Later he has the board with white wheels again.
When Alex walks home after writing "Paranoid Park" in his notebook, the notebook disappears (in the shot from behind) and then reappears.
At 39 minutes, Detective Lu addresses the group of skaters and refers to those who "rose your hand." Rose is the past participle of rise, not raise (which would be "raised").