[RPG] Can an unconscious character use a healing surge

If the barbarian has no healing surges, she cannot use Second Wind. It requires her to spend one, which she does not have. And since the "Regain Hit Points" clause of Healing and Dying specifies that you have none left, this rule would never apply.

Second Wind^[DDI] states:

You can dig into your resolve and endurance to find an extra burst of vitality. In game terms, you spend a healing surge to regain some of your lost hit points, and you focus on defending yourself. Unless otherwise noted in the statistics block of a monster or a nonplayer character, this action is available only to player characters.

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No, something must trigger the use of a healing surge

In order to use a healing surge, something has to trigger it. The main triggers are going to be Second Wind and healing powers. There isn't a way to just "spend a healing surge".

Note that the Heal skill can allow you to use a Second Wind without spending an action to do so, but it does not allow you to spend a healing surge (without having your Second Wind available).

You can also use Potions to spend a healing surge in order to heal yourself, by spending a minor action to drink, and getting a static value back based on the potion.