Lots:
Head Gasket
Brakes
Clutch Pipe
Overheating
Exhaust fell off
But given the cars age and how I tend to drive it I can't complain...
If you are looking for a cheap car, you can use reliably all year around and covering long distances you are in the wrong place. If however you are after something to have fun in, buy one now. I have never had so much fun driving any other car. People who have modern cars with far more performance still find driving these cars amazing fun.
Seriously I know people do use these cars daily and they can be very reliable, but be warned if you want one you must be willing to spend time or money maintaining the car. Personally I have enjoyed the experience of learning about cars from carrying out most the work myself.
While Midgets can be used for long distances they aren't the most comfortable or petrol efficient of cars, also in winter the cars aren't ideal.
Having re-read this I realise some of it is quite negative, but I think that's cause I want to be realistic to anyone thinking of buying one. They are great cars to own and drive when they are working and the sun is out.
1977 MG Midget = 30 year old toy!