A rescued animal can be as good a pet as any other. The only reason one would pose the question is because they assume all "rescued" pets are from abusive homes; this is not always the case. Many shelter cats and dogs come to be there because their owners are no longer able to take care of them, whether it be due to old age, illness, loss of income. Sometimes it's a simple matter of people having to give their pets up because of building ordinances.

The words stray and mutt have negative connotations, but the animals they refer to are often the most loyal and affectionate that you'll ever find. If you're under the impression that all strays are wild and untrained, it may be that you're actually thinking of feral cats and dogs, animals that were born and raised on the street. Strays are displaced house pets, displaced in that they found themselves out on the street when their owners moved or got tired of them. Sometimes an animal will escape from the house and be unable to find their way back home, thus becoming a sad statistic.  

So you know, even a feral animal can be trained, so long as you're willing to take the time. Building trust is a slow process that requires patience and, above all, compassion, though it's best to have experience working with animals as well. If you shy away from the thought of bringing an abused animal into your home, stop for a minute to ask yourself if you have similar reservations where abused children are concerned. Abuse is not the victim's fault and it is not right to stigmatize them for acts beyond their control. 

Pets available at shelters are frequently mutts, or mixed breeds as I prefer to call them, but, often times, you will also find an assortment of pure bred animals and be able to purchase them for well under the going price.  The great thing about shelter pets is that they are litter box trained; they have been tested for diseases; they are vaccinated and neutered or spayed.  You will also frequently be given information about their temperaments and behavior.  Animals known to have bitten or scratched someone will not be put up for adoption until their behavioral problems subside. Animals known for not being social with other animals will be listed as such. There's no reason they can't thrive in a single pet home. 

On the other hand, animals available at non-franchise pet shops often exist under deplorable conditions. In many instances they have been bred right there is the store or purchased cheaply from puppy mills. Money is usually the store's primary objective. The bare minimum of medical attention is given to the animal. Tender loving care is usually neglected all together.  Forget about valid information about the pet's personality; they'll say anything to make a sale.

Years ago, I purchased a kitten from a small pet shop. She was lethargic but, also, the most beautiful cat I had ever seen.  A tortoiseshell cat, she looked like a work of living art. One half of her face was black, the other half was tan. I paid $70 for her because I wanted to save her from the gross conditions they were keeping her in. The kittens had wet food stuck to their fur and, consequently, flies landing on them. She lived two months in and out of the hospital with a terminal disease contracted from living in unsanitary conditions. For this reason, I urge you to make your local shelter your first stop when looking for a furry friend to add to your household. And, please, report any animal abuse that you witness and any pet shop that operates under shady, unsanitary conditions.