Conversion to single speed: crankset specs?

You'll obviously want to select the appropriate chainring size and crank arm length.

Cranks have a specific chainline (the nominal distance from the chain to the mid-line of the frame). The required chainline is affected by the spacing of the rear wheel dropouts, i.e. a 148mm spaced disc brake MTB frame has a wider chainline than a 130mm spaced rim brake road frame.

If you want to re-use the bottom bracket you'll need a crank with a square taper interface, however there is an annoying issue with matching cranks where the axle is built into the bottom bracket (i.e. '3-piece' crank/BBs: square taper, Octalink etc.): different crank models require a specific bottom bracket axle length to achieve the correct chainline.

Normally, if you can't find a crank that requires the same length axle you have, you'll need to replace the bottom bracket along with the crank. Often, replacing the bottom bracket opens up the possibility of upgrading to a '2-piece' crank with external threaded bottom bracket bearings such as the Shimano Hollowtech II system.

If you are converting a derailleur bike to a single speed (your question is a bit ambiguous as to whether you mean a proper single speed or 'one-by' with a single chainring a derailleur/cassette in the rear) then you can play around with the position of the sprocket on the freehub body to get good chain alignment and chainline isn't important.