I was reading your budget as £20 for a case _and_ a power supply - i.e. £10 each. Maybe I read that wrong and you meant £20 each. Doesn't change my advice since every single no-name PSU I've ever had is now in the bin and every single Antec/Enermax is still going or has been outpaced by power hungry components. I've also learnt from sliced fingers that getting a case that's been designed rather than thrown together is much more useful than getting blood all over everything.
I did mean together, a bundle £20 for midi case with a 500W PSU. You must have been really unlucky as my machines (mostly desktops) have all had regular use and cheap components have never caused me a headache. Just out of interest, what sort of numbers of failed budget PSU's are you talking about?
As to cases, I've always gone with the cheapest and I know what you mean about sharp edges, but I haven't cut myself yet. I do agree that cheap cases can be a little frustrating with lids that don't align correctly, sides that are flimsy and can vibrate a lot. However after build, I rarely find the need to go back inside so I'm prepared to be careful during assembly and then forget about the cheap case for the next 18 to 24 months until it's upgrade time. I always rate functionality above aesthetics with almost everything in life, so if I can cut the case cost to get a faster processor or more ram, then I will always choose to do that. If you head down the aesthetic route too far you'll end up with a mac!
I do have to admit superstition of my own, which is that I always buy ASUS motherboards if I have the choice. I've had several in the past and not one has given me any problems so I've stuck with them. I know this is ridiculous, but that's the way the human brain works, spotting patterns that are not really there based on tiny samples.