My mother suddenly fainted, then went rigid and lost bladder control and urinated immediately. This has now happened thrice. What is this?

Possible seizure. The description you gave sounds like a seizure. You should take your mom to a doctor as soon as possible to determine the cause.
Seizure! Fainting, rigidity and bladder control loss point towards seizure disorder. There are many types. Your mother should see a neurologist for good history, thorough physical examination, diagnostic tests and treatment, sooner the better. Thanks.
Syncope or seizure. Syncope, often called fainting, implies a loss of conciousness even if brief. That could trigger seizure like activity or the entire event could be a seizure as loss of bladder control is common to see as well as brief confusion after event. It is critical that she be seen to determine the cause of the fainting in order to treat underlying issues and prevent again.