On Air Rifles

The law makes no distinction between air rifles and more powerful guns for which you need a licence – they are all classed as firearms. This means that any offence can carry a very heavy penalty.

In a public place the rifle should be in a gun cover and always unloaded.

Going on to private land, or water without permission is trespassing, and if you are carrying an air rifle it becomes armed trespass. Whether the gun is loaded or not is irrelevant – armed trespass is a serious criminal offence carrying heavy penalties.
It is an offence to fire an air rifle pellet beyond the land where you have permission to shoot, unless the occupier of the neighbouring land has also given permission.
It is also against the law to fire an air rifle within 50 feet of the centre of a highway if this results in someone being injured, interrupted or endangered. These offences could be committed when someone is shooting in their garden close to a road and the pellets ricochet onto the highway.
As long as the shooter is complying with firearms law, they can shoot certain pest bird species. These are covered by open general licences which means they can shoot the birds listed, provided they have the landowner’s permission and provided they are doing it for one of the reasons allowed by the licence.
These reasons are to protect crops, to protect game and wildlife and/or to protect public health or safety
Birds that are covered by the open general licences - crows, rooks, jackdaws, magpies, jays, woodpigeon, collared doves and feral pigeons.
Mammal pests can be shot at any time provided you have the landowner’s permission. Air rifles are deemed suitable (by BACS) for brown rats, grey squirrels, stoats, mink and rabbits.

Source - BACS