My friend was having problems with her dog, it had terrible separation anxiety and was completey destroying her house and it had a lot of behavioural problems which were becoming a worry as she has 2 kids (it had been a rescue dog and she had it for 3 or 4 years). But when she tried to arrange an agency to rehome it, everyone refused. She tried the SSPCA and Dog's Trust as well as a couple of more local rescue centres, but everyone refused to take it. She ended up dropping it off at a Police Station and pretending it was a stray she had found. She felt absolutely terrible about doing things this way and she was terrified the police would track her down, but she couldn't cope with it anymore.