Nancy,
It's really up to you. Personally, I would leave all three shoots. If later you don't like it, you can always cut and re-root them.

A pic would be great...this sounds like an interesting plant.

-R

Cactus:

The thorns are SHARP and will even go through leather gloves when mature - I use kitchen tongs when handling.

All I know about it is that the tag said hardy to zone 5.

Here are pics of the cactus - first showing rooted cutting with new branching growth (close up of the twisting trunk inset) The second is the roots that are re-sprouting. Looks like a fourth shoot is starting to grow.

Would like to know what variety this cactus is:

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Most likely an Opuntia...?

I know very little about cactus although I also have a hardy opuntia (paddle cactus) - this one is columnar cactus about 2" in diameter (round with ridges / twists) and does not have pads. It has many branching arms though. If all opuntia have pads, this is not an opuntia - although the new growth looks like the opuntia new growth.

A Cylindropuntia or Austrocylindropuntia. I don't know enough about them to say what your plant is, but I suspect it is the latter genus, they are often cold-hardy plants with long leaves.

Thank you! That gives me information to search.