Mine would be,
1. Naayak The Leader (2013 - Telugu) *ing Ram Charan in a double role, Kajal Aggarwal & Amala Paul.
Hated the blunderful court scene. Both the lookalikes are brought to the court where both of them claiming to be Naayak. Simple thing, conduct a DNA test, the sister of Naayak is present there, the parents of the lookalike too are present in the court. But no, the writers are way too intelligent so the court orders both the lookalikes to be kept in one house along with their family people & let the family people decide who is Naayak based on the behaviour. How did I survive this film?
2. Iddarammayilatho (2013 - Telugu) *ing Allu Arjun, Amala Paul & Catherine Tresa.
So, someone is in coma for a long time & that person has some vital information regarding the villain since he's the brother of the villain. All you've to do is hit him like hell, if he doesn't wake up from coma, light his arm on fire. Due to the shock, the patient will come out of coma & within seconds starts giving all the important information. Where does all these things happen? Yes, in a hospital. Where's the security & doctors? Ah well, the entire family of hero & heroine are killed in the "hospital" in the flashback so that's not a big blunder.
I am not considering the Hollywood films of 2013 because they're plenty starting from The Last Stand till The Hunger Games 2, I can go on. I would neglect Irandaam Ulagam's illogical scenes where a fiat car is all needed for travelling from one planet to another & everyone in the another planet speaks fluent Tamil, in the end the hero travels to third planet & even there he finds another form of heroine who speaks fluent Tamil. But since it's a love story it's fair enough to be illogical.
The power plant scene from Boss is supposed to be a PJ so it doesn't qualify.