Here are the definitions:
A clumsy, inept person, similar to a klutz (also a Yiddish word). The kind of person who always spills his soup.
Someone with constant bad luck. When the shlemiel spills his soup, he probably spills it on the shlimazel.
I am not sure which one was Laverne and which was Shirley. Shirley had pretty good luck. Carmine was a cutie. So maybe she was the shlimazel. And though the girls used a great amount of physical comedy, I remember it being Laverne ending up more than not upside down with her panties showing. Which would make her the shlemiel. But she also had milk and coke dumped over her head more than once... hmmmm.
How do you remember it?