Client was lone parent and asked Job Centre for advice on Tax Credits etc were she to take a job. This all happened last winter.
She was given incorrect advice (not sure of its exact nature as yet), that indicated her Tax Credits would be higher than they actually were.
She based her financial arrangements on having more money than she actually turned out to have. She is now behind with CT.
what are the chances of getting either compensation for an inefficient service or an ex-gratia payment?

You need to look at this: http://www.dwp.gov.uk/publications/dwp/2003/frm/

It's the guidance document that DWP use for assessing compensation claims. It's actually quite useful - evidenced by the fact that they wouldn't tell anyone about it for years!!

This isn't 'official', but I've heard that compensation claims are treated less sympathetically if the client complains to MP first, so use the compensation procedure first, via a complaint. Client can still complain to MP (or Ombudsman) afterwards.

Don't expect much. I only managed to get £50 (many years ago) once when the DCS wrote to my client, who had a child in residential education, saying "we are sorry to hear of the death of your child".... When I phoned up Blackpool, all they could say was "oh..".

What other horrifying experiences are out there?

Yes, it is definitely worth trying for compensation. A year ago one of my clients got almost £5,000 compensation for erroneous advice from the DWP regarding an IS claim. He had won at a tribunal and was encouraged by the chair to go for an ex-gratia payment to cover an earlier period. I have got a claim for another client in at present, but have been told that it could take 6+ months. DWP Maladministration Manual noted in previous posting is essential reading. Good luck!