Pick the screw head (socket) clean and clear of dirt and rust.
A handheld tiny drill-bit will do. Make sure its deep enough to fully accommodate your screwdriver tip.
Try a NEW screwdriver (or new tip, if interchangeable) of the right size. Tips do wear, as they're supposed to, and a new one has so much better contact and torque.
Squirt a drop of WD-40 or Penetrating Oil where the Screw and Axle meet. Liquid Wrench was previously mentioned.
Fit Mole-grips or a spanner to the screwdriver's shaft for added turning force, while pressing down hard with your other hand.
Heat the axle side with a heated-up soldering iron. Careful not to damage the plastic wheel. Allow to cool, and repeat. This may 'shock' the joint apart inside.
Use an electric drill & screwdriver tip. Hammer action on, and set to the LOWEST (1) torque setting. This will act like an impact driver to shake/shock the seized joint whilst turning with minimal effect, so it doesn't strip the head or repeatedly jump out.
Buy a screw-extractor set from an ironmongers or motorists shop. They usually come in 4 or 5 different sizes. A screw-extractor will cut into the head and then grip like crazy. You'll need a replacement screw, but the axle will be intact.
Finally, when its out, clean then oil the inside thread on the axle. Otherwise your screw will quickly seize again.