If a character was a TRAIT which says may use super strength, picks up and object, next turn may the character not use "may" super strength and dropping the object?

Like Thor and Odin?

Thor can use Super Strength
Odin can use Super Strength.

I believe once you start using it to carry an object you cannot choose to stop using it.

Well i know the power says it can't be countered once you're holding an object... I would guess that, no, you couldn't decide you weren't using it.

No, you cannot drop an object once it is picked up and held.

2011 rulebook pg.19:

HOLDING AN OBJECT
A character can hold only one object at a time.
A character can’t drop an object once it is held.
An object held by a character can’t normally
be targeted, destroyed, or taken away from the
holding character. To dispose of a held object, the
character must use it in an attack. If, however, a
character holding an object is defeated or loses
the ability to use Super Strength, the character
automatically drops the object,


The only way to drop an object is to actually lose Super Strength. Not using is not the same as losing. So even if there was a trait or special power that said "X may use Super Strength" opting to not use it wouldn't let you drop the object.

Super Strength says no such thing. You can counter SS even if the character is holding an object.

But if you opt not to use SS will that prevent you from not using the object in close combat?

No. Read VanisherPunisher's quote from the rules above. Once you pick up that object, you HAVE to use it on you next close combat attack, unless something specifically allows you not to.

I think the only power that bypasses this rule is Quake, and even then, only if you have more than one opposing figure adjacent.

You could choose to not use super strength while holding an object.

But all that happens is you don't get the +2 to your damage when you target a wall, object, or blocking terrain feature.

Actually, there's an exception to your exception. Gorgon can keep the object even if he only has one target for Quake.

TRANSFORMED BY THE MISTS: Gorgon can use Quake and Super Strength. When Gorgon uses Super Strength, he can pick up only standard objects, but they are not required to be used as part of a close combat attack when Gorgon uses Quake.

Are we still talking about the hypothetical "may use super strength" wording?
Because standard PAC super strength doesn't have any optionality to it.

Ultron-13 can Mind Control you with a close combat attack while holding an object *without* having to bash you over the head with it too.

Though the image of bat-sh*t crazy Ultron-13 mind controlling people by hitting them over the head with a rock would be hillarious!

I believe that's actually Spider Jerusalem's special power.

THE CHAIR LEG OF TRUTH: When Spider Jerusalem makes a successful close combat attack while holding an object, after actions resolve he may use mind control as a free action against the same target.

Disclaimer: The above is not a real power, or a real Heroclix character. But it'd be awesome.

Gorilla Grodd can do that too. Also, you can Incapacitate a foe without using the object as long as you don't have Stunning Blow.