What sort of transformers are needed for LED spotlights (MR16)?
My questions are... Is it normal to have one transformer per bulb?


Yes, though not always. Track fittings in particular use many bulbs (usually 12V MR16) off a single transformer/driver.


  My assumption is that the LED bulbs flash because the transformer is not the right one, is this correct and which one should I get?
  
  I also assume the other fitting do not light, either due to faulty wiring or more likey the transformer has blown?


The transformer may have a minimum load, below which it won't switch on or will oscillate between on and off. Have you tried putting the old halogens back in? If they work, the transformer is OK.


  Do the transformers have to be dimmable, considering we don't have a dimmer switch.


Nope.

If you want LED bulbs (and don't mind about dimmability), it may be easiest to replace the MR16 downlights with mains-voltage GU10 ones, and use GU10 bulbs. No transformer/driver required.