Keep anything that will prove how much you contributed to a Roth IRA each year or converted into a Roth IRA each year. Keep any pre-2011 purchase records from taxable accounts to prove your cost basis when you sell. Keep your tax returns and any related documentation for several years (I think even in cases of suspected fraud they're unlikely to go back more than seven). Keep records of your home purchase price and cost of any major improvements.
Credit card statements and utility bills are generally garbage after they've been paid. Records from closed accounts are usually garbage as well, unless they're documentation of Roth contributions/conversions, establishing your taxable cost basis for shares that you still own, or are related to tax returns that are still within the potential audit window.