The RAF have been named "crabs" due to the fact that they have no limit to the number of sideways paces they can do as a drill movement. While army and navy have a fixed limit of sideways paces the RAF can actually march the entire width of a parade ground sideways.
Another more plausible explanation as to why the RN refer to the RAF as 'crabs' goes back to the days of rum, sodomy and the lash (about last week in fact). One of the more unsavoury aspects of the average matelot was his unfortunate habit of contracting pubic lice or "crabs" during his shore leave. The treatment for this condition was to get a chum or shipmate to apply a liberal application of a greasy blue/grey ointment (known affectionately as "crab fat") to the affected area. The proper name for the ointment was Blue Unction.
With the RN's usual powers of wit and sophistication the RAF were thereafter referred to as 'crab fats' (or crabs for short) as their blue/grey uniform was exactly the same colour as the stuff that the filthy little ratings rubbed on their swollen and lice-ravaged goolies.