"Shooting an air gun is not an arrestable offence. The police are very unlikely to arrest anybody unless the offence carries a five year prison sentence or if the police think the offender may not respond to a summons."
All offences are now arrestable, regardless of prison sentences. Even littering. The Police do, however, have to justify each and every arrest and state why it was necessary, and why it couldn't have been dealt with in another way. The most common reason for arrest is so that evidence can be obtained by way of questioning (interviewing). As a result, the majority of 'offenders' are dealt with by way of arrest.