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Spinning rigid bodies are stable about their axes of smallest and largest moments of inertia. When there are energy dissipation modes, such as bending and propellant slosh, only the largest moment of inertia axis is stable, because rotating about that axis corresponds to least kinetic energy. This is why the Explorer 1's spin stabilization failed (the radio antennae flexed to dissipate energy).

However, rockets that are spin-stabilized are also rotating about their axis of smallest moment of inertia (they are "minor axis spinners"). Since rockets bend and their propellant sloshes, shouldn't this rotation be unstable? My question is whether the reason why spin stabilization works for a rocket's attitude is because the flight time is much less than the time it would take for the rotational instability to grow? Or do rockets use active control systems to stabilize the minor spin axis?

Note that an answer to a previous question seems to indicate that spin-stabilization would be unstable for a rocket - however there's no explanation as to how this problem is dealt with.

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Most spin-stabilized rockets are solid-fueled, so slosh-free. Solid fueled stages also tend to have heavier structure than liquid-fueled ones, because the propellant container needs to contain combustion pressure, so they are less bendy than a liquid stage of similar volume would be. The proportions of spin-stabilized solid upper stages tend to be fairly squat, e.g. Star 48, which further reduces longitudinal flexing (and maybe has a moment-of-inertia benefit as well?).

Spin stabilization has been used for liquid stages over short durations, as well, though sloshing instabilities have caused problems in such cases.

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    $\begingroup$ Additionally, it is true that the burns of such stages tend to be short. The Star 48 motors only burn for around a minute and a half at most. $\endgroup$ – Christopher James Huff Jul 18 '20 at 1:43
  • $\begingroup$ @Russell Borogove Do you have any thoughts on this sciencedirect.com/science/article/pii/S1000936117301449 - I've expanded a bit more in my answer. $\endgroup$ – Puffin Jul 20 '20 at 19:03
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    $\begingroup$ Nope. My mental model shuts down when "precession" and "nutation" enter the conversation. $\endgroup$ – Russell Borogove Jul 20 '20 at 21:56
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Note that Explorer 1's spin stabilization worked fine for getting it into orbit. Dissipation during the few minutes of thrust was not enough to cause trouble.

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A dynamically unstable spinner can be operated with Active Nutation Damping. This "simply" means using a control loop to conduct small manoeuvres to reduce precession. The direction of the thrust is in the body spin-axis and is at a finite radius from the spin vector (the further out the more effect per pulse).

Obviously it needs quite precise timing so that the pulses occur at the right time. The designer needs to choose the right sensor and processing chain to determine the phase with respect to the spin rate and another sensor to detect the nutation angle; so that the pulse size can be chosen to progressively reduce as the nutation is brought under control.

I was surprised to see this https://www.sciencedirect.com/science/article/pii/S1000936117301449 which suggests spin stabilised solid upper stages have often had unanticipated nutation problems; I must confess I've only glanced at it but it seems to relate to movement of the hot gas inside the solid motor.

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