I am a single parent with a child that is almost 2. I was never married to my child's father and we have never gone to court for custody. He has signed the birth certificate. He has not seen my child in 10 months nor attempted to. We have tried to locate him through child support 3 times and have been unsuccessful each time. He will sometimes contact me from time to time asking about my child but that is as far as its gone. What are my chances of filing for child abandonment in Florida and receiving full rights?

You file for custody, you can use abandonment as one of the issues. And you can ( as it appears you already have" primary custody"

This does not mean that the father can not have or file years in the future for visitation. You can not take his rights away if that is what you think. But you can have custody both physical and legal. Even if he does not visit for several years, if he has a change of heart, he can come latter and ask for visits. How and when would be up to the court.