There are two reasons to spin a satellite. One is spin stabilization during launch vehicle operation, which reduces the amount of off-trajectory deviation to a specified coning angle and reduce the offset mass effect of slosh of liquid propellant or slag. The other is to induce a “barbeque roll” that evenly distributes solar radiation across the body of the satellite to prevent it from becoming excessively or unequally heated, and provide long term gyroscopic stability.
For spin stabilization, this is typically done with solid motors that remain attached after action time. Because solids will continue to provide a small amount of residual thrust after the primary burning “action time” due to burning up any remaining slivers of propellant (tail-off) and outgassing of the elastomeric liner from heat being emitted by molten aluminum residue from the propellant (slag), it is often desired to hang onto the motor after action time in order to prevent it from running into the satellite or space vehicle. The preferred solution from a strictly controls point of view is to thrust terminate the motor; that is, either fire retro motors that push it back away from the satellite after separation, or blow open the forward end so that it can’t provide net thrust. However, this can produce undesirable debris and residue that might damage the space vehicle or interfere with its sensors, and so the motor is often retained until the thrust is low enough that it won’t run into the space vehicle. When this is done, it is necessary to keep the vehicle from being pushed off course by the residual thrust. The low and uneven level of thrust makes normal thrust vectoring control (TVC) from the nozzle (either by a gimballed nozzle or a fixed nozzle with liquid injection TVC) unusable for control, and thus requires either an external attitude control system (ACS) or restricting how much the thrust can push the vehicle off course by constantly reorienting the off-axis thrust (spinning). Much of this is due to the unequal distribution of slag mass, so spinning also helps to keep that material pinned to the side and smeared somewhat evenly around the base. This type of maneuver is generally done about the minimum axis of rotational inertia, and once the satellite is fully deployed, some kind of retro-roll motor or a flywheel is used to slow or stop the roll motion as it will not be stable long term and may interfere with the other functions of the vehicle like observation or communication.
The barbeque roll maneuver, on the other hand, is typically done about the maximum axis of rotational inertia (either naturally or the space vehicle will deploy arms or tethered weights). The roll rate for this is slower and is intended to provide some degree of gyroscopic stability against any tumbling motions, and to maintain the satellite in thermal equilibrium. For instance, on the Apollo capsules, a slow (nearly imperceivable) roll was intermittently done in order to keep the lightly insulated capsule from overheating in the constant sunlight.
Once you start an object spinning in vacuum of space where there is essentially no drag to slow it down, it will keep spinning practically forever until you induce torque (via a reaction wheel or roll control thrusters). Even a slight amount of unintended and uncontrolled roll can be a significant control problem.
Just a pedantic point of note, but this isn’t actually an astrophysics question. Astrophysics is the study of the natural physical behavior of stars and other celestial bodies like planets, galaxies, et cetera. This query is more of a space vehicle dynamics and orbital mechanics question.