are consumer grade power supplies designed to run 24/7?

Look for power supplies that are built to specifications set forth by Underwriters Laboratories (UL).

Power supplies certified by UL are almost certainly guaranteed not to burn your house down.

I'm sure they are fire safe. They may give up more quickly than server grad power supplies though, so if you're running it 24/7 try to get one with two or three years warranty.

(As an anectode, when I had a PC power supply give (after about 14 months) up and brought it to the dealer and asked if they normally die that quick, he said "those (meaning the specific brand) at least survive the first year. power supplies and hard drives are like tyres on your car, they are consumed by being used.")

I've run 6 cheap desktop machines 24/7 for years, and I never had any problems.

However, I did lose one machine to a power surge.

There's always a small risk of fire, but UL rated equipment is at least tested and found to meet a decent (minimal) level of risk. Wisest course of action is to only have them on when you really need them on. If that means 24/7, don't worry about it too much though.

I run here a little server, read simple desktop with server OS, for a long time. I had no problems so far. Only on summer days I sometimes shutdown my server, because it is to hot in my room. This server has a standard psu. Every 3 or 4 month I shutdown my server and clean it from the inside, so there is never to much dust inside the server.