Hang a paper towel holder under a cabinet without drilling holes?

Just purchase a vertical holder that stands on your counter. Something like this . . . Sold almost everywhere.

https://www.amazon.com/dp/B01LZ4JFSQ/ref=asc_df_B01LZ4JFSQ5066203/?tag=hyprod-20&creative=395033&creativeASIN=B01LZ4JFSQ&linkCode=df0&hvadid=194859264681&hvpos=1o13&hvnetw=g&hvrand=16971533058924110020&hvpone=&hvptwo=&hvqmt=&hvdev=t&hvdvcmdl=&hvlocint=&hvlocphy=9031180&hvtargid=pla-315180499391

Velcro works great! You can use adhesive-backed Velcro; the male strip attached to the underside of the cupboard, the female strip attached to the paper towel holder. When' you're ready to remove it, aim a hair dryer at the pieces and peel the Velcro off.

Command tape should work too.

That's a problem. I can think of things, but none that will work without screwing in the brackets for it. They do sell paper towel holders that sit on the counter top.

Yes they do, https://www.walmart.com/search/?query=paper%20towel%20holders%20for%20countertop&typeahead=paper%20towel%20holder

I saw one on line the other that slips onto a shelf. Look on Amazon, Overstock, Walter Drake, Lillian Vernon...you'll eventually find it.

You can buy 3M Command strips. You can remove them at any time with no damage to the underlying surface.

http://www.homedepot.com/s/command%20strips?NCNI-5