1. Fail to file a return for any tax year that one is due on your worldwide income? The statute of limitations nevers runs out to assess taxes for that year.
2. Fail to pay taxes on past filed returns or assessed by the IRS? The normal statute of limitations is 10 years from the date of assessment and filing a tax lien (that may be a later date from the date you filed the return) to collect tax. However, if you leave the country or the tax is assessed while you are outside the USA, that statue of limitations is put on hold until you return to the USA when it starts to run again.
3. Failed to file Foreign Bank Account Reporting Forms (FBAR or now form 114)? The statute of limitations to assess penalties for failing to file is 6 years from the due date of each years forms.