I think it is Gaea Foundation. It was awesome :D But it would be strange for me, to have box with cosmetics always with me, when i am a guy.

I think that Balzac would be useful, but it depends on the users potential.

In my opinion the most useful Teigus would be:

Spectator, read peoples minds

Mastema, never need to walk again.

Lionelle, healing factor.

Perfector if you are a doctor, writter or artist.

without particullar order.

About Gaea Foundation, the teigu always disappears when it is use, so if you use the teigu before leaving your house or use it without anyone looking, no one will find it weird 

controlling water would have it's advantages, as would creating ice, you'd never have to walk again. perfector would definitely have it's advantages, balzac could be very useful, shambhala would allow you to go anywhere in seconds. susanoo would help out a ton, with his ability to do household chores so well. i guess there isn't a definitive most useful imperial arms 

Demon's Extract, Gaea Foundation, Murasame, Lionelle, Extase, Pumpkin, Hekatonkheires (Koro), and Primus Imperator

Also, if the Imperial Arms actually existed, they'd either be used exclusively by the Japanese empire, or be used by people working for governments or the military. Hell, we wouldn't even know about their existence anyways as they'd be top secret.