who solves the cases?

I don't think he ever shot anyone. Lol

They rarely use a weapon. Once in a while, Mr. Steele socks the bad guy. Laura solves more cases than anyone. But, she's a trained detective.

The impression I get is that Laura solves (or at least brings all the clues together); but often Brosnan has some out-of-the-blue insight (usually having something to do with movies) that provides a crucial piece of the puzzle. Hope this helps.

Laura is the detective but Steele usually has a eureka moment- even if he doesn't know it- that helps her solve the case. The first season Murphy was the partner more or less and Steele was more of a comic relief. Mildred also helps solve cases especially in season 4. The show was supposed to be a vehicle for Zimbalist brosnan just got lucky by being cast as Steele. I always missed Murphy and Bernice and thought they should have been kept