Longtime knife guy, my 2 pennies: most of mine I run dry. If it's getting sticky or gritty, it needs to be cleaned well with dish detergent and very hot running water.
If you felt like it needs it, a drop of whatever you have handy is fine. Any machine oil, I used to use RemOil, mineral oil for food use. Just rinse and wipe down after heavy use. Food/organic matter gets dish detergent and water, alcohol (ethanol wipes are super handy for this away from home, have multiple uses, isopropyl works in a pinch).
I usually leave carbon steel naked if it's being used often enough. Stuff in storage that is rust prone gets Renaissance Wax.