You give notice as soon as you have a non-contingent offer from your prospective new employer. Not a moment sooner.
You tell your prospective new employer that you cannot give notice to your current employer until you have a non-contingent offer. Also tell them what notice period you intend to allow.
Use the same communication method you have been using so far; e-mail, voicemail, telegram, whatever.
After that, you can pester the hiring manager at your prospective new employer for progress reports. Let him pester his HR dept.
EDIT: John Feltz is correct. The time scale for clearing these kinds of contingencies is a matter of days, usually around five to fifteen business days for driving, drug, and credit.
A true background check takes months, but only the federal government and their biggest contractors do that. Most employers contract with a "security consultant" who essentially just looks you up on Google and then waits five business days before reporting the search results.
MORE EDIT: Of course when you ask about whether you have cleared yet, you must not give the impression that you are anxiously waiting to learn if you can be cleared. Emphasize the "yet" part of your question, and act as if the result itself is a foregone conclusion.
If you are worried that your manner of asking may arouse suspicions, then don't ask. You'd merely be trading one anxiety for another.