Some recent and common ones:
The BT 'Engineer' Scam:
Typically you'll get a call from a supposed BT engineer concerning your broadband router, and that they'll need to undertake some work / reconfigure it to prevent loss of internet services. This is just an attempt to get you (and them) logged into your PC using remote software, which will give them access to your PC and files, where they will no doubt try and access files or try to get personal information or other details. BT never call in this manner, so a few choice words and put the phone down.
The 'Your PC is infected' pop up message on your PC whilst browsing the web:
This one can pop up when browsing and may play an audio file giving some dire warning about your PC being infected with a virus etc. The following is a typical screenshot. Clicking anywhere on the page can make it appear Full Screen, at which point they put a fake website element onto the screen so it looks like you are not on Microsoft's own support site as can be seen with the red rectangle highlighted:
The number they try and get you to call is of course all a fake call centre and they will invariably get you to install some remote access software, at which point they will open a command line and type in a few DOS commands that simply run directory listings and pretend they are running a scan. Then they will run some NET STAT commands and show a load of crap to try and convince you that there is a virus and that you have malware - pointing to a load of web addresses of course.. They then open up notepad and start offer you 'support packages' that they type out as you go along
Of course the aim is to get you to cough up the cash either by giving them your card details, redirecting you to a scam payment site, or by getting you to buy iTunes vouchers

.
I would suggest you close full screen view (where it will generally reveal the fake website address), or press ALT and F4 to close the browser, then clear the history and cache of your browser. But in either case it's all a scam. Vulnerable people have lost big bucks this way.