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.
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.