This is a quick example showing how Final Cut Pro's SmoothCam filter can benefit the outcome for some of your hand-held shots that originally have just too much shake. Although very nice for some shots like this, the filter is not for everyone. It cuts out a good portion of the original frame from principle so make sure to shoot wide if you know you want to use this feature. It also has a pretty motorized look which might not fit the aesthetic theme that you're aiming for. I happen to like it a lot for walking/standing shots that move around a lot like this.
Shot stock hand-held on an HVX200 with no additional filters, stabilizers, or plugins used.