Does the neutral wire float or go to the neutral bus on a sub panel?
No. The grounded (neutral) wire should not float.
In a 120/240V single split phase system, a subpanel should be wired similar to this.
Main Service Panel
Notice that in the main panel the grounded (neutral), and grounding buses are bonded (electrically connected). This means that grounded (neutral), and grounding conductors of a branch circuit can be connected to either bus.
In a subpanel, the grounded (neutral) and grounding buses are not bonded. Which means grounded (neutral) and grounding conductors of branch circuits, must terminate at the appropriate bus.
If the subpanel is installed in a separate building, it might be supplied with a 3 or 4 wire cable, and there may or may not be a separate grounding electrode system installed. There are numerous factors that determine how a subpanel in a separate building is wired, which is beyond the scope of this answer.
The subpanel's neutral bus must be tied to the neutral on the master panel. Neutrals should never be allowed to float. That always causes problems down the road.
The only weirdity of a subpanel is that the grounds are a separate bus (apart from the neutrals) and that is independently connected to the main panel's ground/neutral bus. There is a screw which connects the bus to ground: For a subpanel, remove that screw.