Having fostered for behavioural, socialisation or medical reasons, maybe 18 'problem' dogs over the years, some for weeks and some for years/to the end of their life, it surprised me how many are wrongly labeled a problem.
As David says, there are lots that just aren't suited to live in kennels, they are trained though and don't have the classic behavioral issues. The problem is, if the Rescue Centre assessment says they can't be handled by staff in the kennels for whatever reason then they're automatically deemed a problem dog, and they're not able to rehome a problem dog or have the resources to work with them. Of all the dogs that have passed through my pack, every one was successfully rehomed (apart from the 3 I fell for completely so much I adopted them myself!).
Santi was a foster dog that I kept. Real problem dog apparently, by around 5 yr old he'd been given up by his owners as they had children and he'd gone to 4 different families through the Centre and they all dumped him back there unable to cope such were the extent of his behavioural problems. To me, he's the perfect dog and I did no training with him, only building trust and his confidence in me. Never need a lead, good with stock, travels anywhere on bus, train, ferry, truck or tractor. His only issue is howling when he's alone, and that is a result of his experience with previous owners.
This is him not long after he arrived, with 2 other dogs I had fostered at the time, Ted the Collie (a real head turner, and Ginty the mongrel, but neither had a problem in my opinion.
Ted was a mess having lived in an angry, emotional setting and just needed to trust again. Ginty was 3x too heavy and had never left his garden before owner went to a care home. Absolutely fantastic dogs, tear in my eye now thinking of what they gave me. It's not the dogs that have issues, it's the people.