Protect Her - S1-E1
Factual error: When Jules and Hannah are talking about the financial shenanigans at Owen's firm, Jules says that the owner of Owen's company had fraudulently driven up the price of the stock before the IPO. This is impossible. The IPO is the first time the shares become tradeable.
Other mistake: Grady opens Owen's phone with the wrong passcode. The code is 082007 but he enters 08207.
Revealing mistake: Hannah sits in a conference room and pulls a landline phone towards her to dial her friend, asking about the will Owen left. The phone however isn't in any way connected, hence no call could have taken place.
Continuity mistake: Hannah enters the bar wearing a University of Texas hooded sweatshirt with its unmistakable school color when she asks Charley to take her to see her father. When they arrive at her father's house, she is wearing a plain dark sweatshirt.