Virgin Mobile will generally unlock any phone for a fee of £15.32, but like Three you can't get a free unlock from it, no matter what your circumstances are.
Your network should be able to unlock any handset for you - even the latest and greatest phones to hit the market. Most of the networks have an online form that you can fill in to get your handset unlocked or alternatively you can call them.
Either way you'll need your IMEI number, so it's worth having that handy. Generally you'll be able to find this on your phone's settings screen (for example on an Android handset you go into 'About phone' and then 'Status'), but if you're having trouble locating it you can also enter *#06# on your dial pad to bring it up.
Do it yourself
If your network isn't willing to unlock your phone (or charges too much for the privilege) then worry not, for there are other options. Before proceeding with any of the following methods though, be aware that getting anyone other than the network or manufacturer to unlock your phone may void your warranty.
In some cases it's entirely possible to unlock a handset yourself free of charge. All some phones need is an unlock code, and certain sites list these. However, generally you'll only find them for older, non-smartphone handsets, and even then the bulk of them are for Nokias. But if you've got an older phone, even something other than a Nokia, then it's worth a look.
Before you start you'll need to know your make, model and IMEI number. The make and model should be easy to find (hint: check the box, the manual or even the phone itself). With them in hand you can head to unlock.nokia.free.org if you've got a Nokia or do a Google search if you've got anything else, though obviously be careful to check that it's a legitimate site if you head elsewhere.
Occasionally do-it-yourself methods do pop up for smartphones. For example the Samsung Galaxy S3 can be unlocked with an app called 'GalaxSim Unlock', which can be downloaded free of charge from Google Play - though you need a rooted phone to use it.
Similarly, depending on what firmware you're on you may be able to unlock a jail-broken iPhone. However, Apple is more resistant to unlocking than most other manufacturers - so in many cases, particularly if you've got a newer model such as the iPhone 5, approaching your network directly will be your best bet.
If all else fails you can always pay someone independent to unlock your phone for you. This can be done both through websites and from high street stores and market stalls, though of course it's worth shopping around and where possible checking the legitimacy of the person or company before proceeding.
The terms, pricing, methods and time frames of these services can vary wildly. Generally if you're getting it unlocked online you simply provide the site with your phone's make, model, IMEI number and the network it's locked to, pay a fee and wait for them to send you an unlock code. This keeps it pretty simple, but you still want to make sure that the site is legit and that you're getting good value for money.
As a starting point, though, Mobile Unlocked uses this method and is a legitimate site with a 100% money back guarantee if for any reason it fails to unlock your phone. It will also unlock major handsets such as the iPhone 5 (albeit for a seriously hefty £74.99 upwards fee) and the Samsung Galaxy Note 2 (for a more reasonable £17.99 upwards).
Alternatively, giffgaff's site has a list of other companies that offer an unlocking service, along with some information on them and user reviews.
Some sites may ask you to post your phone to them, but this is generally best avoided because it adds time, risk and hassle to the process and shouldn't normally be necessary.
If you take your phone to the high street, methods sometimes vary, with unlockers potentially physically attaching a cable to your phone to get it unlocked. This can work, though it can also run the risk of damaging your phone.
It's always best to make sure the person or store offering the service is legitimate before handing your phone over, so a physical shop is a better bet than a market stall and you should read reviews or find feedback about them where possible.
One way or another you should now have the information you need to get your phone unlocked, so go out and enjoy the freedom to choose a new network, or go nuts and get SIM cards for them all. The world is your oyster.