Kaz and Mick are being careful - and I'm with them on this.
In basic terms ....
You need to be able to keep (and produce) records to complete a tax return if you need to do one, or if the Inland Revenue want to check up on you. Tax law says that as an individual, you need to keep records for a minimum two years (call it three, effectively).
Now if you've worked for a firm for the whole tax year, most people think just keeping the End-of-Year P60 is fine. If you've changed employment during the year, then keeping the P45 from the old firm and the P60 from the new. In most cases, it probably is ......
But I would recommend you be slightly cautious and wait until you've got your P60 before you shred your payslips. Sometimes firms are dilatory in issuing P60s, depends on the efficiency of the payroll department.
And another reason for keeping payslips is - what happens if you get paid expenses along with your pay? Expenses can be paid tax-free - and don't figure on your P60. If you've destroyed your payslips, you've then lost the link between your P60 and what's going to be credited on your bank statements. Along comes the Inland Revenue and demands your bank statements .......
Of course it may "never happen to you" but the Inland Revenue can and do look at people's affairs and the rules regarding Child Benefits etc. are dragging more and more people into the "self-assessment / complete a tax return" net. Loads of adverts around at the moment advising higher earners to register by 05/10/13 or else; you've probably seen them.
So bottom line - "let's be careful out there" - keep all financial records for the allotted time, and don't start shredding until you're sure you're beyond it.
Further reading for individuals:
http://www.hmrc.gov.uk/sa/record-keeping.htmJust my 2c worth as a retired number-cruncher ...