Continuity mistake: Between the pilot episode and "The Terror" (which was never shown on TV), the costumes of Batmanuel and Captain Liberty change substantially even though there is no more than an hour between the events of the episodes.
Continuity mistake: When The Tick first enters Arthur's apartment he breaks the door which then rests at a sharp angle. Then later when The Tick breaks the light fixture thinking it is the 'secret trigger' the broken door is propped up straight. In all shots after that one the door is at a sharp angle again.
The Terror - S1-E9
Revealing mistake: When Captain Liberty writes The Terror's phone number on the napkin we can plainly see that she is using a blue bic ballpoint pen - but when we get to see the napkin she had written the number on (again - second napkin) it was written by a felt-tip marker.
Continuity mistake: When we first see Captain Liberty's dog with the grenade in its mouth, the handle is at the top. Then the scene cuts and when we return the grenade handle is now at the side. Another cut and when we return before the dog barks the grenade handle is at the top again.