Air rifles are perfectly legal and pretty cheap and would dispatch of a rat quite efficiently. I grew up in the country and they were used for said purpose on a regular basis.
I would however check with the police about the legality of letting off pellets in an urban environment. I would be very interested to know as if they approve I shall be off to the Anchor Bankside to dispatch of those bloody pigeons!
You can buy glue traps for rats and mice...works like giant flypaper...we had a terrible mice problem, I did try the traps, but found the sight of them being stuck very upsetting, and then had the problem of how to dispatch them ....first tried dropping them into a bucket of water after getting drunk enough to ignore the bubbles... then gave up and bought electronic pest repellant for seventy pounds.....it worked....
Having read it I presume that the youngster I saw the other day was breaking several laws. He was about 16, and on a bicycle, with a friend - cycling on the pavement opposite OKR tesco. What looked like a pistol, fell from his pocket onto the ground - it made a very plastic hollow sound as it hit the ground which made me think it was a replica. he quickly picked it up and put it into his side pocket of his hooded top, and the two of them sped off. As he passed me, he put his hand into said pocket, and raised it towards me, as if to threaten me, as I was the only person around, and rode off.
should probably have informed the local constabulary, but don't know were the nearest one is, and am sure that they wouldn't have done anything about it anyway.
it seems like shooting rats is probably illegal in these circumstances (though I'd love to have a go!)