I have had 11 AXs in the past 18 years. Two 1.1 petrols, seven 1.0 petrols, and two diesels. The diesels are head and shoulders better than the petrol versions, especially on the motorway cruising at speed; far quieter and up to 20 MPG better. It's possible to get 75 MPG from the diesel, but 67 to 69 is more realistic.
The petrol car can be thrown around more, as the weight of the diesel engine does load up the steering when pushed. The 1.0 is the pick of the petrol engines if economy is important, although the 1.1 is pretty quick by anyone's standard, but is very noisy and the MPG can drop to low 30s, especially in the winter months. The 1.0 pretty much gives around 50 MPG in general use, just squeezing around 60 on a very gentle motorway run; 3 to 4 MPG less in the winter though.
They all rust just behind the headlights on the top plates under the bonnet. Apart from that, the little AX is no worse than any other car. I have never had a break down ever. The only problem I had was an alternator went on one of my 1.0 petrols, but it had covered over 100000 miles, so it's fair enough.
I do all the servicing myself; even the cam belt change on the diesel only takes about an hour, and it's always worth checking the water pump when changing the belt, as the belt drives the pump, and for about £20 and 2 bolts, it's a no brainer.
The drive shafts have a reputation for being a bit fragile, but are easy to change; just keep an eye on the outer CV boots, as they split and can ruin the joint very quickly. The boots can be a pain to change, and for £25 for the whole drive shaft on eBay, you might as well do the whole thing.
If you are over 6 foot and challenged in the weight department, like my 23 stone mass, forget the 5 door, as you could sell tickets to watch me trying to get in and out of one, but the 3 door cars have very large doors, and are fine apart from that.
Don't be put off with buying one of these cars; just remember there is not much to go wrong, but there is no power steering, and it can be hard to park unless you are very strong.
My latest one is a 1996 diesel with 61000 miles. I don't think it's been looked after too well, as it still had the original air filter in it. I have serviced it, changed the cam belt, and have covered 9300 miles in three months going backwards and forward to London where I work. The car has averaged 69.8 MPG and been faultless; enough said.