85 MR2

Bad grounds, faulty switch, loose-fitting fuse.

If they all go dark at the same time, it has to be a wiring issue before any junction to the branch harness wires. Loose fitting to the fuse or a loose broken wire in or out of the fuse box on that circuit is the first place to look. Also check same around any light relays if so equipped.

Any idea where it's grounded? Also, would the headlights and running lights be on the same ground?

Could be but probably not. Ground is the "common" side of the electrical system. All grounds eventually work back to the negative battery terminal through many individual wires that typically use the frame or engine block to attach to. They, in turn, connect to the negative battery lead which also connects to the frame or block. If you are still having problems, look for ground connections that appear loose or corroded, a typical problem source for this type of difficulty (and sometimes hard to find).

Your most likely suspect on an '85 MR2 is going to be the light switch itself. It's a bit of strange design, they have these multiple contact points in the switch with what looks like a BB that is spring-loaded and moves over the contact points and that's probably not making good contact - they tend to get kind of gunked up. You can take the switch apart (carefully) and clean the contacts and metal ball, this may work for you. If you do the cleanup method, buy yourself some di-electric grease and put some of that on the contact points and ball to help lubricate things and also hopefully help ensure the contacts stay clean. Another option is this...I found my parking lights worked consistently on my MR2 but when I had it switched to headlights they could occasionally go off on me when driving down the road (kind of unpleasant to say the least). I determined which wire was the parking light wire by operating the switch with it removed so that I could use a test light to identify which wire received power only when parking lights were selected, then I determined which wire was the headlight wire, by again using test light to identify which wire received power when headlights were selected. I then bought a couple of T-tap wire connectors, crimped one on the parking light wire and one on the headlight wire, then attached a male spade connector to each end of a short piece of 18 gauge wire and connected that to the parking light and headlight wires that I previously T-tapped. I jumped the power from the parking system to my headlight circuit eliminating the lousy headlight connection in the switch. The downfall of my "fix" is that I can't have just the parking lights on, but since I virtually never use only parking lights, it served the purpose. If the switch checks out fine, then I'd start inspecting the wiring itself and using a test light on different sections of the wiring for the headlight circuit until you can identify where the issue is (you can test for ground or power depending upon what you have the test light clipped to). Maybe a critter chewed on your wiring somewhere or you have a worn section from vibration, but my gut tells me it's going to be that switch.