You need to check the laws of your town but there are usually the same ones for each place. Now I know this might be a long post but so little know about their rights and rules of leafleting.
Usually you are fine if the place you are leafleting is a public place.
Don’t attach the leaflets anywhere you don’t have permission to and don’t trespass then you can get done for burglary believe it or not as this offence can be committed by entering as a trespasser with intent to commit criminal damage.
You can still leaflet in what is called “quasi-public” places, where the public has an implied right to enter – for example, a supermarket or a petrol station forecourt. Again you are not trespassing in this situation until you are actually asked to leave and refuse to do so.
Check the content of your leaflets as it is seen as an offence to display any writing, sign or other visible representation which is threatening, abusive or insulting within the sight or hearing of a person likely to be caused harassment, alarm or distress. Now pictures of animal rights may not seem to you as being “threatening” or etc but the police or security might so bear it in mind.
The police might occasionally threaten you with arrest for breach of the peace for handing out leaflets. However you could only be arrested in this situation where the contents of the leaflet were inciting violence or provoking the use of violence by others.
Just think public places and be respectful and you shouldn’t have a problem.