[RPG] In which situations would the ability to prevent creature healing be most useful

There are a number of different strategies one can take (besides the possibly obvious: remove material spell components/focuses/holy symbols/etc.):

1. Make the jail special
   

• Suppress sound (e.g., permanent or ongoing silence) to eliminate verbal spell components

• Suppress light so that the spell caster cannot, for example, target individuals or locations via line-of-sight (e.g., use illusions over doors or windows to block line of sight).

• Suppress divination (e.g., permanent or ongoing nondetection) so that the spell caster’s buddies cannot scry in order to pop in via, say, teleport, and rescue them.

• The jail is built in an area that "naturally" suppresses, hampers or distorts magic.

• The jail is constructed using permanent or ongoing anti-magic spells

NOTE: Following up on one of SevenSidedDie’s comments on the original question, spell caster-proof jails may be rare enough and expensive enough things, that one would not expect small towns, or even necessarily any particular city to have the capacity to hold a spell caster (although this will depend somewhat on how common spell casters are in the campaign setting, for example, if every small town has a few high level casters, then these kinds of jail enhancements might be common). This kind of scarcity might make the possession of a spell caster-proof jail a special resource for a region, or even between nations/principalities/city states/etc. where extradition treaties, rendition treaties, or even trade treaties are negotiated around arranging the incarceration of a spell caster (imagine a side narrative about arranging the transfer of a incarcerated spell caster: fun!).

2. Alter the incarcerated spell caster
   

• Magically compel the spell caster not to use spells while incarcerated in jail (e.g., a geas, some really nasty custom curse, mark of justice, etc.).

• Make the spell caster wear some difficult to remove (probably homebrew) magic item that suppresses somatic movements, sounds, or even magic.

• Tag the incarcerated caster with difficult to remove tracking and summoning magic, and make them aware that there will be sanctions for jailbreaks.
  

• Place the spell caster in some kind of suspended animation (e.g., the vanish sink spell of 1st edition AD&D).

• Drug the spell caster so that they are incapable of casting (per orlp’s excellent comment).

3. Have the jail include guardians who are practiced at containing spell casters.
   

• Abjurationist wardens

• Mage-slayer wardens (perhaps equipped for and specializing in non-lethal combat and restraint)

• Wardens equipped with anti-magic devices

• Wardens that are magic-suppressing, magic resistant, or magic immune monsters or constructs.

4. Make compelling appeals to the incarcerated spell caster’s reason, morality or emotions

• Threaten the spell caster’s familiar, family, colleagues, or benefactor with sanctions (e.g., your family pays a huge fine if you jail break; your familiar will be destroyed if you jail break; your teacher will be stripped of standing in the college of magic if vouchsafed spell casting students jail break; your first born child will receive the spell caster’s sentence if they jail break; etc.).

• Appeal to the sanctity of law (if that is something that the spell caster cares about).

• Escalate the legal threats against the spell caster by compounding an original crime with a jail break (e.g., the punishment originally was 6 weeks in the clink and a small fine, now it is the amputation of your right hand).

• Make plea deals, so that publicly the spell caster is seen as punished, but the powers that be arrange to trade benefit for leniency in actual sentencing.

Mostly balanced except for Dance of the Dead

Overall this subclass doesn't appear to be too unbalanced. Playtesting it as you are probably gives you a better idea than any of us would have purely from reading the rules.

Grim Secrets

Part one of this feature seems fine. Though no other college gain a cantrip at this level I don't feel this is fundamentally different to any of the others.

Part two is harder to guess at as it grants an ongoing improvement rather than a static benefit. To fully know if this is OP I would have to compare all necromancy spells to all bard spells and see if this adds a significant amount of power. I feel as though it may be a little over powered compared to the existing subclasses. Potentially consider moving this to the sixth level to mirror Additional Magical Secrets of the Lore Bard.

Haunted Eyes

Advantage against fear is common enough from many sources so not a big deal. Necrotic resistance is good but situational so not game breaking.

Dance of the Dead

This feature seems to be both over powered and goes against the normal design principles. For reference the other subclasses gain at this level:

• Additional Magical Secrets (Lore), two spells of your choice
• Extra Attack (Valor/Swords), as it says
• Mantle of Majesty (Glamour), Command as bonus action for 1 minute. Once per long rest
• Fool's Insight (Satire), detect thought CHA mod per long rest
• Mantle of Whispers (Whispers), situational disguise

This feature also uses Bardic Inspiration in a way that no other subclass uses them, more akin to the way superiority die are used than bardic inspiration.

This is the feature that needs the most work and I would consider scrapping entirely for something different.

Rule the still heart

This feature is fine, it could actually be considered weak since it only affects one creature per long rest and they get a saving (possibly with advantage).

Suggestions

This isn't too bad as a first attempt of the subclass. I would suggest re-doing dance of the dead. Potentially replacing it entirely. Allow me to propose some features for you, not sure if it suits 6th or 14th level better though.

Grave Humour

When an Undead Creature under your control, that you can see within 60 feet of you makes an attack roll, an ability check or saving throw, you can expend one of your uses of Bardic Inspiration, rolling a Bardic Inspiration die and adding the number rolled to the Undead's result. You must decide before the DM says whether the roll succeeds or fails.

Puppet Master - credit to Doc in comments

When you use the Attack action on your turn, if a creature you control can see you, it can use its reaction to make a melee attack. Only one creature you control can do this per round.