Usually Monstera will have some root buds (for the development of aeriel roots) coming off of the bottoms of some of the stems. I would pick those segments to try to propagate. I have a "monster" monstera (excuse the pun) in a 35 gallon container that puts out tons of long roots from the stems. This plant is huge, its leaf stems are 5 ft long and leaves are the size of platters. It would be nothing to cut them and start them growing. SMaller plants, I don't know.

My monstera's "vines" are fairly thick, maybe like a really fat asparagus. Is that the vine to cut and propagate? It does have lots of arial roots. Thanks for the help!

To be honest, just lopping off any part that has nice root buds should be fine.

I sometimes chop the stem into 3 inch pieces, let them harden off for a day or two, then lay with nodes down on top of the soil. You don't even need leaves to do this.

You can also cut the stems into any size from 3 inch and up, with the leaves on, and stick the end into soil, making sure a node or two is below the soil.

These things will practically propagate themselves.

I always cut off the air roots when I propagate.

I forgot to say --

I also root Monstera in water all the time.

I kept five or six thick branches with large leaves in a big plastic jug of water for almost a year.