I occasionally see the bobbies on the beat out here in the sticks. They don't ride bicycles or motorbikes though, instead preferring to ride horses. Its quite nice as you can clearly see them and they are happy for us citizens to come up and say hello to the horses etc.
The one thing that does amuse me though is all the signs up saying it is an offence to NOT pick up after you dog has done a poo. But you do find horse poo on the streets here and there. Now I've not seen a Police Horse do a poo on the street yet, and there are other horses out in the sticks here.... but the thought pondered my mind as to whether the Police have to "arrest" themselves for allowing their horses to foul the streets