I located this which has a summary of some Do's / Dont's of some of the 39 Melachot It says:
Whitening/Scouring What: Cleansing in any manor of absorbent materials with absorbed dirt or the bleaching of such materials.
Do: Wash dishes, cups, etc. Use paper napkins to clean up spills Dry
dishes with a cloth Hang up a suit jacket Hang up a rain coat Dry ones
hands on a towel ( (דרך לכלוך Spray static guard on a clingy dress
Walk around in the rain
Don't: Wash, soak or scrub dirty materials Scrub or rub hard a
material even dry ( (שפשוף Soak dirty towels ( (שרייה Ring out water
from a white towel ( (סחיטה Put water on a stain on clothes or carpet
Polish silver Have a water balloon fight Use towels to soak up spills
Put a wet garment near a radiator to dry off Hang up clothes to dry
that are typically washed (ie. not dry cleaned)
You have a separate question regarding spraying, which is addressed here:
Air fresheners and perfumes that are sprayed are not considered to be
Zoreh, and therefore it is permissable use them on Shabbat. However,
one may not spray perfume onto clothing or fabric. Directly onto skin
is okay.
From what I read about static guard, I gather that they do make them in sheets as opposed to a spray. That may avoid any melachot, that way. Also, you may want to check with a Rav. Perhaps since the scent is secondary and not the intent of using the static guard, it may be permissible. The above quote seems to imply specifically to perfumes on dresses, i.e. - products that are specifically used for scents.