The pet you're attempting to research on must have energy in order to carry out the research. If your pet does have energy and you're still unable to do any research, then there may be a glitch. However, if the pet does not have any energy, you cannot research. I'm not sure on the technical aspect, but I believe 1 point of energy allows you to research 1 gene. I'll be studying this further unless it's confirmed by someone else
Edit:
It's possible that pets that are busy cannot have their genes researched.
"You know what the issue is with this world? Everyone wants a magical solution for their problem, and everyone refuses to believe in magic."
- Jefferson/The Mad Hatter, Once Upon a Time