Clubs can be good but really only if you live close enough to use them frequently I think. I've known people who've been members of the American the British clubs and they seem like quite nice places to hang out for the afternoon/evening and they also do the special things from back home that you don't really get anywhere else in Singapore like Thanksgiving for the American Club and Guy Fawkes Night for the Brit Club, so if you join one connected to your home country it can be good for that.
I think some of them also have a minimum spend per month so that's another reason to live fairly close, so that you keep going. They're especially useful if you live in a house as they have the facilities that you'd get in a condo
(pool, gym, tennis etc), but if you already live in a condo you might find that you don't use the club as much because of that.
I've been to the Brit and the American clubs for lunch (members can sign people in at certain times) and they didn't seem too stuffy. I think the Tanglin club is posher and I heard that the Island Country Club is one of the most exclusive on the island (someone told me you have to get voted in, but I don't know how true that is)
Dog parks - the only ones I know of are in Katong Park on the East Coast and one in West Coast park, but I haven't got a dog so I'm not very clued up on the parks - there might be more of them. If you don't have a car, you have to pre-book a cab apparently to take your dog in it as Muslim cab drivers won't carry dogs, so living near an open space/park would be an advantage so you don't have to keep booking cabs.