Well firstly avoid Gumtree (or any other similar "service") unless you know exactly what you are looking for.
Secondly, £60 is not going to get you much - but you could do quite well, although it is always worth looking new too - it's amazing what you can find.
What you need to do is decide what sort of bike you want.
For commuting there are 5 types to choose from.
A road bike - 700c wheels, generally high geared, light, few options for mounting mudguards/racks etc... often have dropped handlebars. Has a very streamlined (head down) riding position.
A mountain bike - 26" wheels, generally lower geared, heavier, few options for mounting mudguards/racks etc... flat or slightly raised handlebars, has a more upright riding position.
A hybrid/trekking/touring bike - 700c/27" wheels, middle geared, generally plent of options for mounting mudguards/racks etc... can have flat, raised or dropped bars and varied riding positions.
A roadster - 700c wheels?, few, if any, gears - generally in an internal gear hub, quite heavy. Racks, fenders (and dynamos) often standard fit. Moustache handlebars and a very upright riding position.
A folding bike - Small wheels (16", 18" or 20" generally, but there are some 26" folders), few gears either rear cassette or gear hub only. Often have mudguards, racks and dynamos. Quite an upright riding position.
I really depends on what you want to do and how often you want to travel.
If you want to use public transport, then there is an extremely high chance for buses that your bike will have to fold if you want to take it with you. I can't say about the trains.
If you want to go for a mountain bike, then get a rigid - if you really must have suspension, just get it on the front - but I really don't think it is worth it. You will want to change the tyres for slicker road tyres though (extra cost).
I picked up a nice Raleigh Parkway 3 speed hub geared folding bike from a local second hand private dealer recently for £60.
If you can up your budget a bit and look at new, here are some options you might want to think about (and more to the point you can go to your nearest Halfords and try most of them in store)
Raleigh Evo 7 Folding Bike - £199.99
Trax TR.1 Rigid Mountain Bike 2013 - £79.99
Trax T700 Hybrid Bike - £99.99
Apollo Fusion Road Bike - £199.99
This one will be out of your price range, but I included it to showcase the type of bike I was talking about.
Pashley Roadster Sovereign Buckingham Mens Classic Bicycle - £695.00