Personally, "grieving" and "mourning" are synonyms to me. They mean they exact same thing.
"I was grieving over my dead cat."
"I was mourning over my dead cat."
"Regret" is something else. It happens when something occurs that you wish had either never happened or you wish did happen.
"I regret that I didn't ask that girl out to dinner." (wishing something had happened that didn't)
"I regret to inform you that you have cancer." (wishing something never happened that did)