'Arthurs' MotorBike refers to the second hand motorbike which Arthur Rudge bought which appears in multiple episodes of the show. It is never brought up in the series exactly where Arthur purchased the bike from but it can be assumed it was from a News Paper advert.
The bike consists of Motorcycle along with a side-car which is frequently seen being used by Olive. Stan refers to the bike as a dam nuisance mainly because there always a problem with it. Known problems include the bike always breaking down, the starter braking, the handles falling off,a broken engine,broken hooter and coming apart in "The Used Combination" after Stan kicks it.
The sidecar is shown to be wobbly and has been seen to come off from the attachments and fall a part even though the parts have been replaced, polished and fixed. Its not long before something starts to go wrong.
One episode reveals that Jack also had a motorbike at one point which Stan helped to fix. This was the only mention of his motorbike and it was never shown.
The sidecar changes throughout the show. In the very first epsiode it appears in (The Used Combination), the sidecar has a glass panel that must be lifted up and down in order to get into it. As the episode is in black and white, it is unsure what colour it is, but it could possibly be red. In most of the episodes that it appears in the sidecar is red with a black roof, but in the episode Nowhere To Go, the sidecar has changed from black to red, but it later returns to red. In the first scene that it appears in, in the first film of the show, the sidecar is red with a black a roof, but in all the other scenes it appears in, the black roof has been taken off. In Mutiny On The Buses, the sidecar has been painted green instead of red, and it is roofless and a glass panel has been put up in front of where the passenger sits. In Holiday On The Buses, the sidecar has been painted black, the glass panel is gone and it is roofless again.