Sorry about the proper spelling all you weird American people!
Stabilisation in LumaFusion is the best thing since sliced bread. Nearly all my shooting is handheld and I use stabilisation on nearly every clip. You can do them one at a time or multi select clips and apply stabilisation as a preset.
Generally I'd say just tap the on/off switch or choose the standard preset and it does an amazing job. For some clips you might need to crank it up to strong. For really shaky clips you can sometimes get even more strength by exporting the stabilised clip, re-importing it and stabilising the stabilised version!
Shutter correction is for rolling shutter warping but it helps with any small jittery movements as well.
Sometimes changing the Scaling Speed in Advanced Scale can also smooth jitters and jerks.
Occasionally a slow rotation effect is apparent after stabilisation. It can be corrected with key frames in the Frame and Fit tab.* You might expect Advanced Strength rotation in the stabilisation tab to fix this but it doesn't.
You can see the stabilising in action with Advanced Scale: tap Fixed, reduce the scale to about 0.7 and watch the clip's edge moving around when you press play.
I haven't found any use for Advanced Strength with handheld shots. It mainly lets you adjust stabilisation independently on the various axes. It could be useful if say you had panned unevenly using a tripod and only needed to smooth the horizontal movements.
I can't really think of any practical use for Advanced Margin which lets you colour around the clip if it has been reduced in scale . You can also set a video as the background so maybe you could do something complicated with a second version.
* When making small adjustments to rotation in the Frame and Fit Editor it's much easier to use the new key pad option to enter values (tap the pencil icon). Just remember that clockwise rotation is negative. It seems wrong to me but apparently it's a mathematical convention!