Had some help with this earlier, but that was attaching a soft cloth to a cone prop. That worked.
Now I have my fairy princess hat, with veil attached and if I apply an animation to my avatar, they dance off into the night with their princess hat floating in space, unattached. 

Is this a weight map issue like attaching the veil?

Thanks!

Further, I found that if I remove the hat up a little and then Attach it to the figure, it does work. If I move the figure the hat moves with it UNTIL I apply a Dancing Animation.  It is as if the hat is no longer parented/attached to figure.

Probably me just not understanding the software.

Well, if you wish to have the hat follow the head, it has to be attached to the head bone.
(you probably just attach it to the character, in which case it is attached to a root bone)

Point to the head when attaching, or select right bone thereafter:





She is very cute BTW

Hi, thanks for helping!

That is Buttercup, from my first book Fairalon. I was using Poser for the images, and did a little animation, but I see more potential with Iclone for animation.
I changed the setting for buttercup and picked the only bone i could see: FacialBone (image 1)

I can move the character and the hat IS attached, until I apply the heidi dancing animation.

Then she dances out of the hat perfectly and the hat stays put. (image 2)
It is as if applying the dancing animation, cancels out the attachment. (Though, if i look, it says it IS attached.)

First, not the Face bone, but Head bone (one above).
But that is not even the problem. The cone is surrounded with red outline meaning you enabled physics for it. Why?
AFAIR you have a veil as soft cloth, which is attached to the cone. There is no reason to enable Physics for the cone.
So go to frame 1, disable physics for the cone, then "Remove Object Animation" by right mouth click on it in view-port (because it now has an animation of its own written to the timeline).
Finally attach to the Head and the cone would follow the head.