Hi,

I attached a cloth object to a cone; a little veil to make a princess hat.
I tried parenting it to the cone, but if I save the Avatar the Veil did not come with it.
I asked the forum, and was helped saying you have to "attach" the veil to the cone for it to be saved.
When I save it as an avatar, the cone and veil are gone. If I attach the cone to the head and the veil to the cone, they are saved.
Great! BUT...
When I animate the Avatar wearing the Cone Veil, the cone moves with the Avatar, attached to the head, the veil stays where it started.
I know it's me, but what am I doing  wrong?

When I animate the Avatar wearing the Cone Veil, the cone moves with the Avatar, attached to the head, the veil stays where it started.
I know it's me, but what am I doing  wrong?                               

It should work without a problem. It seems you have baked veil animation and then disabled Soft Cloth altogether? Is that possible?
Make sure you have a check box on Soft Cloth globally in Project settings. Remove Vertex Animation from Veil (given soft cloth is enabled for it as well).
And try again.
Here is an improvise setup. Do you have something similar? (it always a good idea to share some screenshots with your setup)

Ahh OK, so you disabled gravity so that it does not fall down, right?

The problem is that you never set any weight map for the cloth. You need to set pinning with it, so that it's static at the pinning points and stay attached.
Default map is a plain white image and you need to changed that. Start from here to understand how cloth weight maps work: https://manual.reallusion.com/iClone_7/ENU/Content/iClone_7/Pro_7.4/20_Physics/Setting_Soft_Weight_with_WeightMap.htm?Highlight=weight

OMG, it worked. That is so cool.
Had to play with the "Weight map" contrast to get it to hang properly, but it appears to be working.
Saved the cone as a prop and the veil was attached and retained the weight map properties, so Woo Hoo!
Thanks so much!