I think almost everyone would agree that the wording is much less clear (at least in intent) than it would be with an explicit "must" or "may", and that it would be better if the 5e designers had been clear one way or the other at least for creatures with 1 action per turn. And better if they'd phrased it in a way that covered the possibility of multiple actions per turn.
There are at least 4 possibilities that are (somewhat) compatible with the rules text wording and/or possible intent. There are valid arguments for either of the first 2 being fully RAW, compatible with the actual text. The third I think not, but can be still a valid option for a DM to pick if they think it makes more sense in their game world / campaign, as with any homebrew or choice to take rules as you think the designers probably intended.
Option 1. Every action that can be used to attempt a save must be used to attempt a save. (It wouldn't force you to use action surge, but if you did, you'd only be able to make another save, letting you use your movement for something else if successful.) @HighDiceRoller's answer makes a reasonable argument for this reading of the text, based on there being no explicit mention of turns for the target creature.
I hope you brought Quickened Dispel Magic if your Wis saves aren't good.
Option 2. (arguably the best fit for the phrasing, and at least a valid one RAW)
One save attempt must be made per turn (costing one action) if you can take an action at all. Any further actions available on your turn can be used for whatever you like, including Action Surge to take the Attack action.
If it was talking about all your actions across all your turns, it should be plural. If you only got one singular save attempt for the whole duration of the spell, that would be quite weird (see option 4). So we interpret this sentence as talking about "on (each of) the dancing creature's turn(s)". The phrasing and the thing it's telling you to do is sufficient to imply that context.
Option 3. (weak consensus that this is not RAW) Attempting a save is something you may spend an action on, if you want. This might have been what the authors meant to say, but doesn't seem compatible with the literal wording in the spell description which isn't optional or conditional.
I'd consider this a houserule that's incompatible with the literal wording of the spell, which is an option except in Adventurer's League play.
This would let spellcasters use their action to cast a spell with no downside if it didn't involve an attack roll (which would be at disadvantage.)
Option 4. (highly unlikely to be intended) You only get one save during the whole duration of the spell. When you attempt it, it costs an action. The previous sentence introduced the context of "While the target is affected by this spell", and then we have singular language for the action and save. This would leave all your other actions free to use however you like.
If we don't assume an "on every turn" in the key sentence, this is a more grammatically compatible reading of the literal text than "on every action" (option 1). I think, it's unlikely to be intended; spells that only allow one save normally allow it when cast, not on the first turn after.
I mostly mention this to point out that not assuming "on every turn" doesn't imply "on every action". @A C commented as much on the option 1 answer, thanks for pointing out that possible reading.
All 4 of these possible rulings have no effect on bonus actions, reactions, or legendary actions: those can all be freely used. Same for hasted actions; as you say, the things you can do with them don't include attempting a save, so you can still make one attack that way (possibly triggering the ability to make a bonus action attack).
5e rules are sometimes written in ways that only cover the common / simple case, like when the creature isn't hasted and isn't using action surge, and isn't incapacitated, so has exactly one action on its turn. In this case, the wording (without a "may") does dictate that the creature spends its only action each turn to attempt a save. Also that movement normally only happens during a creature's turn (or after readying movement as an action).
It's up to the DM to extrapolate from rules written to cover a limited subset of possible situations.
Under these simple-case assumptions, most of the paragraph containing this sentence is talking about constraints on each of the target creature's turns. The use of singular follows from that assumption of only having one action per turn. (This is one way to argue for option 1, that the singular in the rules text would become plural "action(s)" if they were written to accommodate action-surge, forcing you to use all your actions.)
A dancing creature must use all its movement to dance without leaving its space and has disadvantage on Dexterity saving throws and attack rolls. While the target is affected by this spell, other creatures have advantage on attack rolls against it. As an action, a dancing creature makes a Wisdom saving throw to regain control of itself. On a successful save, the spell ends.
An implicit "on its turn" in the "as an action" sentence makes the most sense to me in terms of why the designers would phrase it that way. If they meant to say every available action must be spent on these saving throws, I'd have expected phrasing more like Turn Undead, saying it can only use its action to attempt to regain control. Or at least plural like "for all its actions".
The middle sentence clearly covers time outside their turn, the whole time "while affected by this spell". So that's a weird thing to throw in the middle. But the final sentence doesn't keep that context, unless you want to read it as the option 4 interpretation. Unfortunately this doesn't help us establish anything about the time-frame for this singular action.
Unlike actions, movement / "all its movement" is a continuous thing. It's also not actually limited to the creature's turn: this shuts down chances to move outside its turn, such as legendary actions or in reaction to Dissonant Whispers or a Maneuvering Attack. (That creature can use its reaction to move up to half its speed without provoking opportunity attacks from the target of your attack.)
Perhaps the enchantment spells Command (1st) or Puppet (1st UA) could override the dance's control of the target's locomotion, although Dance is a much higher level spell unless you upcast Command... But that's out of scope for this question.
I think even an effect like Turn Undead or Turn the Unholy (some undead and some fiends aren't immune to charm) couldn't override the dance effect which limits what they can do with their movement. ... must spend its turns trying to move as far away from you as it can. Interestingly, those effects also limit what one can do with an action: For its action, it can use only the Dash action or try to escape from an effect that prevents it from moving. Attempting the save against Irresistible Dance would count as "try to escape", because the effect does otherwise stop it from moving. (Dash increases your movement allowance by your speed; it's not a separate way to use your speed without counting as the "movement" part of your turn.)
Being incapacitated would mean you don't have an action to spend on it, so you don't attempt to save. Tasha's Mind Whip (2nd) is an interesting rules-interaction: "... on its next turn, it must choose whether it gets a move, an action, or a bonus action; it gets only one of the three". Choosing movement would be a total waste (could only use it to dance), and I think is not required. The same way a fear effect doesn't force you to use Action Surge when it requires you to spend your actions to dash. You could choose to get a bonus action, but if you choose to take an action, it can only be to attempt the save. (Assuming an option 1 or option 2 ruling.)