Can I wash my chinstrap with hand soap?

Just wondering, or will it damage the fabric, do I have to use laundry detergent?

Use whatever (a) gets it clean, and (b) gets along with your skin.
I use Dawn, the clear stuff, with minimal scent; sometimes liquid Method unscented.
Some folks use baby shampoo, others Dr Bronner's.
Not necessary to go out and buy something new--use whatever you already have.
You might want to use something that rinses out really well, to avoid soap residue irritating your skin.
Your skin matters more than the fabric--it needs to last a lifetime.

chunkyfrog wrote:You might want to use something that rinses out really well

That would be my concern. Some hand soap might not rinse out of fabrics easily. I use All Free & Clear laundry detergent, which is already on the shelf for the rest of the laundry. It is unscented.