Half terrier, and the terrier half hunts
The Staffordshire Bull Terrier was built in England by crossing bulldogs with terriers, and the two halves still divide the breed's instincts between them. The bulldog side contributes the grip and the famous devotion to people. The terrier side contributes the hunt: terriers were kept to pursue rats and other vermin into tight places and finish the job quickly, and that pursuit instinct sits just under the surface in most Staffies. A fleeing animal flips the switch. The dog that was ambling beside you locks on, accelerates, and if it catches its quarry, tends to hold and shake in the classic terrier pattern rather than simply chasing for the fun of the run.
Prey drive is only one of three separate traits
Adopters often blur three different things into one worry, and with this breed it pays to separate them. Prey drive is about small, fleeing animals. Tolerance of other dogs is a different trait, shaped heavily by early socialisation and individual maturity. Sociability with people is a third trait again, and it is the one Staffies are celebrated for: this is a breed that craves human company and physical closeness. A Staffy can be intense about squirrels, easy with the dog next door, and devoted to every human it meets, all at once. Judging the individual in front of you beats judging the label, because the spread between individual dogs is wide.
Reading the specific dog before you commit
If you are adopting, especially an adult from a rescue, gather evidence rather than impressions. Ask what the dog has lived with before, whether it has caught wildlife on walks, and how it reacts to cats seen through a fence or window. On a trial walk, watch the body language: a soft glance at a pigeon is unremarkable, while a fixed stare, a lowered stalking posture and a trembling refusal to look away tell you the drive is strong and easily triggered. Introductions to resident cats or small pets should happen through a gate or door first, over days, and a dog with an unknown history should never simply be released into a house with free-roaming small animals on day one.
Living safely alongside smaller animals
Management, not hope, is what keeps small animals safe in a Staffy home. Gardens need sound fencing, walks in wildlife-rich areas are best on a lead or long line until recall has been honestly proofed, and small pets like rabbits or guinea pigs live behind doors and latches the dog cannot reach, never loose in the same room unsupervised. Many Staffies do settle into genuine peace with the household cat they were raised beside, and that peace is real, but it is built on familiarity with that individual animal, not on a general promise about cats. In KinsLe, Transferrable Profiles keeps that hard-won knowledge attached to the dog: when a Staffy moves to a new home, its history with cats, wildlife and other animals passes to the new owner intact instead of being lost.