I have tried different grades of fuel before and saw no difference. Once or twice a year I'll fill up my car with Shell's 93 octane ONLY because it has more conditioners and cleaners in it compared to our 87. Our engines aren't designed to run on 89 or above nor do they run as hot to run on it at either peak performance or most efficiently.
The higher you go up in octane the more "stable" the fuel is and the higher temp it needs to ignite (fully). This is why in most luxury/performance/turbo'd cars tell you that you need premium fuel because it is designed to give a certain level of performance. If you put lower grade gas into a car designed to run on premium you will see decreased performance and/or mileage. This is due to premature detonation of gas, also called "knocking". The cars ECU (I think, in conjunction with knock sensor) has to retune engine timing to readjust for proper gas detonation timing and you will see the decreases there.
Since our cars aren't designed for it and they aren't reaching the temps required to detonate all the fuel you should in theory be running a little rich and may even see more soot from your exhaust ports (provided you did this all the time) over running regular. Since you aren't burning all of it off you're going to get more gunk out of it. Doing this on accident won't hurt anything obviously but I'm not sure what the long term effect may be.
I know a lot of people around here know this above but I'm just throwing it out in case anyone stops by and reads and may not know. This is in no way meant to be condescending to anyone. I only mention the above because in my mind that's my thought process of I don't see how we can get more mileage, but I only ever tried premium as opposed to regular, I never used mid. Maybe we have that tiny bit of extra heat that mid could use but not enough for premium?
Unfortunately my daily use is so inconsistent (traffic) that I'm not sure I'd be able to reliably tell you when I would get more. I only know the premium because I used it dedicated on a trip once, a bit more expensive but I was curious. Maybe I should do the same with mid sometime. Also, there should be a few older posts around here about this very topic.