You've got the Nevada State Police, The Clark County Sheriff's Department and the Las Vegas Police Department to choose from in 1957.
The CCPD/LVPD seem to have worn worn a tan coloured shirt with khaki tie and khaki trousers with a tan stripe, sometimes a khaki shirt with tan tie with tan trousers with a khaki stripe. Sometimes tan trousers with a tan shirt and khaki tie. A khaki waist jacket was often worn. Cap was tan with brown leather peak. I'm guessing the khaki jacket and either trouser was a 'winter' uniform.
Nevada State police appear to have worn dark blue uniforms with a light blue stripe down the outer leg. The shirt was a blue/grey colour with royal(?) blue epaulettes and tie. Either dark blue caps or a dark blue montana hat were worn. There seems to be a variation with blue grey trousers with a dark blue stripe.
This pic is from the 60's or 70's...
Here's the badge, but this appears to be a recent one compared to the one the trooper above is wearing. Colours are reversed, but the badge is the same.
I can't say I'm 100% on the above, but I don't think I'll be far off.