Effing Internet Explorer! I just wrote a HUGE long post then lost it!
Starting again!
I'm a volunteer for a small independent rescue and I also work with several others as an independent network rescuer. I have a fair idea of the big ones but we don't tend to have much to do with them.
Firstly - ANY and ALL responsible rescues will want to know that you have provision for your dog lined up before adoption. So, you will need to look into employing a reputable dog walker or sitter, or finding a helpful pal or family member. Seek the former via personal recommendation, your vet, breed clubs, local rescue, agility club, trainers etc and check all references as you would for a childminder.
Secondly, ALL decent rescues will:
Homecheck - for YOUR wellbeing, happiness and safety as well as the dogs
Vaccinate
Neuter (unless against vet's advice if, say, the dog is epileptic or similar)
Microchip
Assess ALL their dogs thoroughly
Offer a lifetimes support and advice
Insist that they meet all the household and that all adults are on board with owning a dog
Be able to answer all your questions and have plenty of their own for you!
Insist by written contract that the dog is returned to them if you cannot keep him... at ANY point in his life.
Please, please, please AVOID any rescue which doesn't do all of these things as a minimum and as standard.
Some of the big ones (Battersea, Manchester, Cheshire, Birmingham "Dogs Homes" and Wood Green Animal Shelters, for example) are not rescues but council pounds. Other pounds also often go by the name of "XYZ Rescue" etc, which is deceptive - so ASK if they run the council's stray contract. Problem is that almost all pounds DON'T offer the above best, safe practice. Some (many, probably most) will just let you walk in, pay your money and walk out with a dog without assessing him, checking you, meeting the whole household and any other pets and they don't care what happens once they have your cash.
Even WGAS and Battersea don't ALWAYS homecheck, which is a bloody disgrace and a potential for injury, unhappiness, the placing of an unsuitable dog, the missing of obvious and avoidable dangers, such as a hole in the owner's fence which causes a dog to escape and get run over and so on. That doesn't mean avoid Battersea or WGAS but INSIST that they homecheck you. Heaven forbid that they don't and you end up with a hurt person or dead dog because of it.
Not too long ago an MNer "adopted" a dog from what she thought was a rescue. When I asked the name I was able to tell her it was a pound... but by then it was too late. She'd come on here for help and advice because the dog had bitten her DS. Although she had had a (literally) 5 minute visit from the local dog warden by way of a homecheck the dog clearly hadn't been properly assessed, if at all, and besides the adoption should NEVER have been allowed to take place. The lady was a novice owner, the dog an adolescent collie (cross lurcher IIRC), the lady was young, had no garden and her SN child (I think I am right in saying he is Autistic) was also often wheelchair-bound.
So, pick your rescue VERY carefully and INSIST that they do it all as I've advised above, please.
Don't be deceived by flashy surroundings either, often independent rescues run on a shoestring are more responsible than the big ones although they look chaotic (trust me, I KNOW these kind of dedicated, responsible, chaotic and impoverished rescues!). Independent rescue also often have less one size fits all policies about rehoming where there are children and assess the situation, dog and family on it's merits.
So, first, find your dog walker/sitter.
Then research your rescues.
Happy to try to suggest one or 2 (can't promise but will try) if you'd like to state your area.
HTH :)