I've lived with cats practically all my life - the last 4 years we've been catless after my lovely she-cat died from kidney complications (the longest we've gone without having a cat), and since then I've had DS2 (now 3.4) and didn't feel able to cope with a new animal in our lives too. DS1 (6) has never forgotten our lovely old cat and reminds me exactly how long it's been since we had a cat in the house. 
Finally, I feel ready to have another cat. DS2 likes cats a great deal and makes happy noises when he sees them, though he generally doesn't notice animals right away and certainly isn't one to chase them around. (DS2 has autism and is non-verbal at the moment.)
So we paid a few visits to local sanctuaries, asking to see animals who were used to children or wouldn't be freaked out by them. DS2's age is no longer an issue, it seems (I rang up when he was 2 and was told he'd be too young) and he and DS1 had a very successful introduction with a beautiful ginger boy-cat who's 3. It took us a couple of visits to decide, and a lot of dithering because it's a big decision, but the cat seemed fine with both boys - affectionate and purry with DS1 and tolerant of DS2! (The cat came from a flat where he was living with a much young family - baby and a 2 year old. By all accounts he was fine with them.)
So we've found the holy grail, a young cat who's used to young children, likes them and us, and we love him. The sanctuary owner was very happy with the introduction in the pen with the boys and said our home check would be just to make sure we were on a suitable road, had somewhere for the litter tray, etc.
She mentioned that the woman doing the check was 'a bit scary, very strict - but lovely'. 
She's coming from a fair distance to do it and I had a bit of an odd phone call with her yesterday. For a start, because I'd been out until that call, and apparently the mobile number she'd been given wasn't being recognised (no idea), she was fairly irate that I was 'impossible to get hold of'. 
I obviously apologised and tried to explain that I'd been in and out all day but she talked over me, and then told me when she couldn't do the visit. Unfortunately the first day she can do the visit, DS1 and I are setting off for a weekend camp. I ended up offering to come back early on the sunday (I do appreciate she's doing this voluntarily) so she could meet all of us. I told her that the cat sanctuary woman had said she was pleased to see the cat respond so well to the boys, and that she'd said this check was just to see the house, but Home Visit Woman said sniffily - well, I always need to see everyone in the family in their home, especially with young children. Fair enough, I suppose - but I can't help feeling a bit doomladen.
DS2, with his ASD, can be a bit ... odd. He is very gentle and as I said, not one to chase or persecute animals. I'm also hoping, since he likes cats, that having a friendly cat in the house might bring him out of his shell and help him develop. I'm also quite sure I can police the cat and DS2, and there are lots of places the cat can get to where DS2 can't follow, as you're generally advised to do with cats and smaller children. But I'm still bricking it a bit. The sniffy Home Visit Woman already seemed a bit peeved with us and I'm worried DS2 will be having a not-great day, and put her off us completely.
So positive stories about home checks with young children, or children with special needs, please! 