Both have pros and cons. You could lose your keys and be locked out of your house (in which the spare key in the house is of no use), or you could lose your purse (in which case you lose both your main key and your spare key). I keep spares in my house and my car. Also good to keep a spare with a friend/family.
I'm a guy who doesn't carry a purse. However, if I did, and it didn't clash with my outfit,I wouldn't put a key there. In the event the purse gets lifted, the crim. would have access to too much information.
Spare key to what? Spare key to car in magnet hideaway on the under-body of the car has saved me a few times. Spare key to house in plant by the front door. I like to have my spares near where I would need them ;)
Spare key to the car: inside the house.Spare key to the house: in a lockbox outside the house.And, to keep me from locking my everyday keys in the trunk after loading groceries: everyday keys--house and car--on a long lanyard attached to a cross-body purse. I only detach the keys when I'm driving or going into/out of the house. (Hopefully this will keep me from making that particular mistake again. It has so far.)