I don't know about the Western world... but in Asia, there's this unspoken law about resignations - You're not suppose to tell people what you're going to do or where you're going. At best you make up an excuse like,
"my mum's sick... or you have a permanent hernia that's leaking bile and crap, so it won't be fair for your office mates to deal with it".
IMHO, it's pure Dark Management bull behaviour. You've probably fucked up so bad, that you're afraid that your current employer will call up your new boss explaining intimate details of the screw up.
Secondly, when you do bump into your old boss; you'd have this weird ass look of someone thinking up another lie explaining why you're with a Vendor which you've signed a 3 year non-compete agreement during your previous employment.
Be a man!
Just own up and tell the whole world that you're leaving your current Gawd forsaken dump for the next dump. Who knows; folks may just think you have balls for a change. Not to mention how mortal enemies now suddenly come right at you and give you a congratulatory pat in the back...
"No hard feelings yah, and good luck"
"Sure, thanks mate", and what you really want to say is... "Fuck you and die!"