The most simple use for a scarf is what it was made for, to keep warm in cold weather.
2-To lower the risk of snow blindness
By cutting a couple of slits into it and wrapping it around the eyes, it can reduce the amount of glare from sunlight on snow or water from entering the eyes,.
3-To pick up something
In artic weather, metal sticks to unprotected skin, but not if there is a scarf between the skin and the metal. It can also be used for a brief time to handle hot things although that might burn it.
4-To swat insects
If flies are being a problem, this can be used to kill them.
5-As a draft excluder
Some places are drafty and shoved into a gap, a scarf can block a draft.
6-To strain water
A scarf can be used to strain algae, mud and the like out of water, although the water will still most likely need to be boiled before it can be drunk safely.
7-To tie someone's hands
If the PCs are kidnapping someone or need to tie someone up to stop them raising the alarm or hurting themselves or others, a scarf can be used.
8-To gag somebody
If the PCs need to stop someone calling for help, a scarf stuffed into their mouth could do the trick.
9-To blindfold somebody
If the PCs need to stop a kidnap victim seeing where they are going, a scarf makes a good blindfold.
10-To strangle someone
A scarf can be wrapped around someone's neck and used to throttle them, and unlike other weapons like swords, knives, clubs or guns, can be legally carried anywhere, even onto a plane. In real life the Thugs carried out their murders with a scarf.
11-To carry secret messages
A message can be put on a scarf that for example, only reveals itself when the scarf has the flame of a candle run close to it.
12-As a mask
If the PCs are committing a crime and do not want witnesses to identify them they can tie scarfs around their faces.
13-To avoid suffocation
If the PCs are in a burning building, they can wrap scarfs around their faces to keep smoke out of their noses and throats.
14-To wipe away fingerprints
If the PCs are committing a crime and do not want to leave their fingerprints around to identify them, they can wipe the fingerprints away with a scarf.
15-As a bandage
If a PC gets a cut, in a sword fight for example, then a scarf can be used as a bandage to prevent loss of blood.
16-To support a broken arm
If a PC has a fall or a fight and breaks an arm, a scarf can be used as an emergency support until the PC can get proper medical attention.
17-As a sling
If the PCs have access to stones, then a scarf can be used to fling them at their foes. The humble sling is one of the oldest weapons in the world.
18-As a symbol
A scarf in a certain colour and worn a certain way might be the secret symbol of a secret society and is used by the members as an identifying mark.
19-As a belt
If somebody's belt has been cut in a sword fight, then a scarf can be used to keep their trousers up.
20-As a signalling device
If the PCs need to send a signal, a scarf can be set on fire and waved around. It might mean the difference between being rescued or not.
21-As a swag bag
A scarf could be tied to a stick and have a few things put inside them to make them easier to carry.
22-To ward off bug bites
In a place full of biting insects, a scarf can be used to keep them away from somebody's face
23-As a dog lead
If the PCs have a dog and want to keep it under control then a scarf can be used as a lead.
24-To drug somebody with
If the PCs have access to a sleeping drug, then they can pour it onto a scarf and use it to drug someone to kidnap them or operate on them.
25-To Mark territory with
In a city de facto ruled by various violent criminal gangs, different coloured scarves are used to mark which gangs control which city blocks. You mess with them at your peril.
26- To help pull someone out of quicksand with
If one or more PCs are together and one of them gets caught in quicksand, a scarf can be used to help pull the stuck PC free.
27-As a booby trap
If the scarf is tied across a door, corridor or path at ankle height it could trip up anybody sneaking around in the dark.
28-To cast a spell with
A scarf could have a spell cast on it that would be cast when the scarf is waved about, and it could be an attack spell, a defence spell or some other kind of spell.
A scarf could have hidden chain mail inside it as covert armour to fend off would be assassins.
30-Add your own use here
Additional Ideas (2)
31. Historical uses have been to prevent chafing around the by soldiers wearing armor and by WW1 aviators to prevent chafing as they constantly swiveled their heads watching for enemy planes.
32. For a handfasting ceremony. This is the wedding ceremony where the priestess ritually ties the two betrothed's hands together while blessing their union.