I use head bands - Nike terry-cloth (?) is my favorite, but I also like my two Halo bands.

BUT I still like to wipe my brow often, so I keep two bandannas in my handlebar bag. Susan, a lovely riding friend, rubber bands a bandanna to her right wrist, so it's always at hand. (Side note: It can be pretty noticeable when she signals a hole in the road!)

Yes, all you naysayers! A HANDLEBAR BAG!

• Access is superb!
• I can carry stuff like gloves, rain jacket, sun screen, extra headbands, alternative eye wear.
• I can also bring stuff home - books, magazines, a loaf of bread. The bag has straps that can allow oversize items to stick out.
• Once I was on a group ride up the seacoast on a HOT day. We stopped for a snack in Maine. They had unlimited refills on good lemonade. I rode away with a 16 ounce glass of lemonade with ice in my bag - I felt like a king! (admittedly a lame king, but still a king)

I'm old and slow, so my advice is often irrelevant.