The phrasal verb to sign off on something means to formally give your approval or support to a decision or course of action. To 'sign off on' something implies that there is something in writing or some kind of evidence which traces the decision back to the person who approved it. This is a phrase a verb you will find used in both social and professional contexts.

1. I need to hold back on signing off on the deal until I can be sure that the necessary funds are in place.
2. We need to wait for the boss to sign off on the project before work can begin.
3. I never signed off on that memo to the staff. You have overreached your authority by sending it.

1. I wish the council with sign off on the permission to use that green space as a playground. My children will be grown up before it is built.
2. I almost have enough money to buy the car I'm just waiting for the bank manager to sign off on my loan.