Lots of opinions so far, and any could be correct. IME (which is just one more possible perspective), the DC - and + terminals are your power supply. The Ground terminal is a safety ground, connected to the ground lug of the mains plug. It likely exists in the power supply to ensure that if the PSU develops a fault and AC mains is present where it shouldn't be, any conductive path to the chassis will cause a short and a blown fuse / breaker.
Giving you a convenient access point is useful if you'd like to do something similar with your load device. I.e., a metal chassis that has no particular potential WRT to the circuit within, but a safe return path for stray voltages, noise, or faults. Normally, the Ground will be galvanically isolated from your circuit. Connecting the - terminal to ground defeats this for probably no good reason -- and could do more harm than good.
For example, in telecom where lots of devices run on -48vDC, there's almost always a separate ground terminal. Connecting that to either PSU terminal could mean seriously destructive current loops through the chassis and equipment racks. It's there for saftey and noise immunity, and should never be connected to the PSU rails.
On most home appliances, a metal chassis will have a connection back to the mains ground lug. However, all internal voltages will be floating with relation to mains ground (which is why you measure no voltage to either terminal -- the circuit is incomplete). This becomes necessary as you inter-connect appliances. For example, a TV and cable box. Each one's internal PSU floats with relation to the AC mains, so they're free to use single-ended positive supplies, dual-rail supplies with a center tap, or whatever they want. There's no inherent coupling between them except for the I/O points -- which could be AC or DC coupled, as appropriate. Still, since all of their chassiseses are at AC ground potential, you can set them on top of each other, or touch them at the same time, and all it does is equalize their safety ground potential. No harm done.