Lots of helpful suggestions from others here, and I'll add two more.

One is to put the camera into continuous shooting mode, rather than single shot mode. Then press the shutter and listen to it click off 3 shots. Often the first shot is just fine, but if I've jiggled the camera while pressing the shutter release, my hand will be steady again during the next two shots and I have two back ups to choose from that are likely to be better. I use this more frequently for a dark interior at fairly wide angle, though I've also used it outdoors. This is somewhat similar to the suggestion made earlier about shooting in High-speed Burst HQ in the Scene mode, but in that case the camera takes over and uses automatic settings if that matters to you.

The second solution is one I use outdoors at extreme zoom, where a secondary problem is that the slightest movement of the camera can dramatically alter the framing of a shot, sometimes cutting off part that you might have wanted to retain. Set the self-timer to 2-second delay. Then when you press the shutter release, you have 2 seconds--a perfect amount of time--to reframe the shot if it got messed up, stop breathing, brace yourself, and let the camera take the shot while you're holding steady. Needless to say, this works better with architectural details than it does with birds in flight.

I have the SX50, but these suggestions would presumably work with any super zoom.

Lots of helpful suggestions from others here, and I'll add two more.

One is to put the camera into continuous shooting mode, rather than single shot mode. Then press the shutter and listen to it click off 3 shots. Often the first shot is just fine, but if I've jiggled the camera while pressing the shutter release, my hand will be steady again during the next two shots and I have two back ups to choose from that are likely to be better. I use this more frequently for a dark interior at fairly wide angle, though I've also used it outdoors. This is somewhat similar to the suggestion made earlier about shooting in High-speed Burst HQ in the Scene mode, but in that case the camera takes over and uses automatic settings if that matters to you.

The second solution is one I use outdoors at extreme zoom, where a secondary problem is that the slightest movement of the camera can dramatically alter the framing of a shot, sometimes cutting off part that you might have wanted to retain. Set the self-timer to 2-second delay. Then when you press the shutter release, you have 2 seconds--a perfect amount of time--to reframe the shot if it got messed up, stop breathing, brace yourself, and let the camera take the shot while you're holding steady. Needless to say, this works better with architectural details than it does with birds in flight.

I have the SX50, but these suggestions would presumably work with any super zoom.