What happens at 0 HP?

The following will happen when you drop to 0 HP (Unless you are purposely knocked unconscious.)

1. You fall prone and drop anything you have in your hands. Note that anyone (friend or foe) can use a free action to grab 1 item.
2. You start making death saving throws on your turn IN PRIVATE. Nobody knows your status unless they do a medicine check on you or simply start healing or trying to stabilize

Some other Options Not implemented but presented here for thoughts:
Home-brew Table:

D12	Injury	Duration
1	Twisted Back. Severe Back pain radiates through your body. You have disadvantage on attack rolls, Athletics checks, and STR and CON saving throws	This condition lasts until you spend 10 minutes without moving or taking any actions and you receive magical healing.
2	Broken Leg. Your movement speed is halved. Whenever you attempt to move in combat, you must make a DC 15 CON save. On a failed save, you cannot move on that turn	This condition lasts until you or someone else spends 1 minute to set the break and you receive magical healing.
3	Broken Arm. You cannot hold anything in your primary hand; you have disadvantage on all attack rolls made with your off-hand; you cannot use weapons that require two hands; and spells you cast that require a somatic component have a 50% chance to fail, though no spell sot is consumed.	This condition lasts until you or someone else spends 1 minute to set the break and you receive magical healing.
4	Armor Damage. If you're wearing armor or using a shield, your AC is reduced by 2.	This condition lasts until you spend 10 minutes repairing your armor.
5	Leg Injury. Your movement speed is halved.	This condition lasts until you receive magical healing.
6	Arm Injury. Whenever you take an action in combat, you must make a DC 10 CON save. CON a failed save, you drop whatever you are holding in your primary hand.	This condition lasts until you receive magical healing.
7	Internal Injury. An internal injury causes excruciating pain when you attempt an action in combat. Whenever you attempt an action in combat you must make a DC 10 CON save. On a failed save, you lose your action.	This condition lasts until you spend 10 minutes without moving or taking any actions and you receive magical healing.
8	Winded. You got the wind knocked out of you. On your turn in combat, you can either move or take an action, but not both.	This condition lasts until you spend an entire round without using your movement or action or until you receive magical healing.
9	Bleeding out. You lose 1d6 HP at the start of each of your turns.	This condition lasts until either you or someone else uses an action to bind the wound or you receive magical healing.
10	Disoriented. The pain of your injuries disorients you to the world around you. Whenever you move in combat, you must make a DC 10 CON save. On a failed save, you move your entire movement speed in a random direction.	This condition lasts 1d4 minutes or until you receive magical healing.
11	Blurred Vision. You have disadvantage on Wisdom (Perception) checks that rely on sight and on ranged attack rolls.	This condition lasts 1d4 minutes or until you receive magical healing.
12	Minor Scar. The scar doesn't have any adverse effect, but 50% of chicks dig it.	This condition lasts 1d4 minutes or until you receive magical healing.

Or directly from DM Guid (But I think these are too much a penalty. Lingering injuries