Consider panning, in sports photography, where you track a player in a field while the background blurs from side to side. Also in motor sports, some nature photography, or any time a photographer wants a motion effect, perhaps in street photography, or even snapshots at family events.

Edit: similar effect in travel photos, where someone shoots from a moving vehicle - far background might look fairly stable, near background (cyclists & vehicles) blur, people in your vehicle might be stable, depending on shutter speed and how much bouncing on the road.