When do I spin up my rocket's second stage in KSP?

You can totally do spin stabilization in KSP.

Usually you won't really want to, though, at least not in a Career / Science game, because by the time the game gives you access to any good ways to spin up your rocket (such as Sepratrons or Breaking Ground DLC robotics) you will have long since unlocked other, more practical ways of stabilization. But if you want to do it for a self-imposed challenge — say, to get into orbit using only solid rockets and no reaction wheels — then you certainly can.

In any case, the thing about spin stabilization is that once your rocket is spinning on its axis, it's going to stay pointed the way it's pointing, at least on average. Any pitch or yaw torque is only going to make the rocket wobble a bit, but the axis of rotation will stay mostly fixed. What that means, in practice, is that you'll want to have your rocket pointing the way you want it to accelerate before you spin it up. And, for an upper stage, that typically means horizontally.

Anyway, assuming that your upper stage is aerodynamically stable while the burned-out lower stage is still attached to it (which it should be, unless your payload is super draggy, since with the lower stage empty of fuel your center of mass should be pretty far front), what I'd recommend is something like the following flight plan:

1. Use the lower stage(s) to lift your apoapsis somewhere between 40 km and 120 km or so, i.e. into upper atmosphere or low space. Don't decouple the previous stage yet. Gravity and aerodynamics should ensure that your rocket will gradually turn horizontal as it approaches its apoapsis. (If your apoapsis is above 70 km, resist the temptation to time warp after leaving the atmosphere, since in unmodded KSP that will stop craft rotation! You can physics warp using Alt+>, though.)
2. Once your rocket is pointing horizontally (or a bit above horizontal, if you're still inside the atmosphere), spin it up using Sepratrons. It should now stay pointed horizontally while it spins.
3. You can now decouple the previous stage and fire the solid-fueled upper stage. If you were already in space and planned your upper stage delta-v right, this should put you in a stable orbit.
4. If you started your upper stage burn while still in the atmosphere, you'll (probably, assuming your upper stage had enough delta-v) instead end up on a suborbital trajectory with a high apoapsis and a periapsis around the altitude where you started the upper stage burn. In that case, after decoupling the spent stage, you'll need to stop your craft from rotating again somehow, turn it around and do one more (hopefully small) burn at apoapsis to lift your periapsis out of the atmosphere. Or I suppose you could instead keep the craft spinning, physics warp until you're close to apoapsis and the craft is pointed roughly retrograde, and then fire a forward-pointing sepratron or two for the final circularization burn.

No, this won't generally be the most efficient possible launch trajectory, and you won't have much fine control over the exact orbit you'll end up in. But that's the price you pay for doing it the hard way.