The Shock of the New (1) - S1-E1
Continuity mistake: Towards the end of this episode, Emma tells Brennan that her powers don't work if she's stressed. The first time they met, earlier in this episode, she is running from the GS agents and very panicked and stressed, yet she easily uses her powers to ask for his help.
Crime of the New Century - S1-E9
Continuity mistake: In the fight scene at the start of the episode, Jesse knocks one of the GS agents into the water - in subsequent shots he has disappeared.
Visible crew/equipment: At the dance club when Caleb Mathias holds Emma, Brennan comes to the rescue with a few swift airborne kicks. During one particularly high kick, the lines attached to Brennan are in perfect view above him.
Revealing mistake: When Shalimar is getting ready to leap on Gabriel, there is a quick glimpse of support wires attached to her.
Continuity mistake: When Andrea jumps off the roof at the end of "Where Evil Dwells", season 3, she is falling face down for the first two shots, then when it switches to a third shot she is falling face up.
Factual error: Each time Adam Kane appears as a hologram, you can hear the sounds of his footsteps. A hologram would not make a sound other than the projected voice.