Does a dies trigger go on the stack if the creature is regenerated instead of dying?

Regeneration replaces the destruction of a permanent. Under CR 701.15b, when a regeneration shield is used, the creature is instead tapped, all damage is removed from it, and it's removed from combat — but it stays on the battlefield. It does not go to the graveyard.

A 'dies' triggered ability uses the template 'when [this creature] dies' and is defined under CR 700.4 as meaning 'is put into a graveyard from the battlefield.' If the creature never reaches the graveyard, this event never occurs, so the trigger condition is never met.

Because the trigger condition is never satisfied, the ability never triggers and nothing is placed on the stack. CR 603.2 specifies that a triggered ability triggers only when its trigger event actually occurs. Regeneration prevents that event from happening entirely.

Concrete example: Your opponent attacks with a 2/2 creature that has 'when this creature dies, draw a card.' You cast Doom Blade targeting it, but your opponent activates a regeneration ability in response. When Doom Blade resolves, the regeneration shield replaces the destruction — the creature stays on the battlefield tapped. No 'dies' trigger goes on the stack, and your opponent draws nothing.

Note: Regeneration only replaces destruction (CR 701.15b). If the creature would die for a different reason — such as being sacrificed or having 0 toughness — regeneration does not apply, the creature does die, and any 'dies' triggers would function normally.