Can an unconscious character use an action (like second wind or a healing power) to spend a healing surge?
Can an unconscious character spend a healing surge due to some external effect?
However, if an unconscious character is the target of an effect such as Healing WordDDI then they can spend the healing surge.
Also, an adjacent character could make a HealDDI skill check to allow an unconscious/dying character to spend his or her second wind.
One way an Unconscious (but not Dying) character can spend a healing surge is through magical healing performed by another PC. Leader healing powers are one means; healing potions poured down the throat are another. If the PC is Dying and gets 20 or more on the death save s/he also pops up (it doesn't have to be a natural 20 if the PC has bonuses to death saves). The only other way is, if by some small chance the dying character hasn't spent his/her second wind yet, an adjacent ally can succeed on a DC10 Heal check to allow the unconscious character to spend it (though without the +2 to defenses).
If the second wind has already been spent, mundane healing applied to a Dying character can only stabilize him/her, which leaves them Unconscious and no longer rolling death saves, so no chance of popping up by beating the DC 20. So in some instances it may be advisable to postpone the Heal check to stabilize. Typically these are: already used up all magical healing, or if the Dying PC has a significant bonus to death saves, leading to both a good chance of the save beating 20 and a low chance of accumulating death strikes.