I'm a father and I have signed the Birth Certificate for my child, me and my ex are not married and never have been, do I have joint custody for my child with my ex?

Basically, yes. But, push come to shove, you would need a court order to enforce it.

Any question on law needs to include your general locale as laws vary by area.

In many areas, parents do not sign the birth certificate. The hospital makes the certificate out and submits it to Vital Records. For legal paternity, you may both have to sign an Acknowledgment of Paternity. But if you are the legal father than you have equal rights with the mother.