IV: HP - ATK - DEF

I would say for a defender all 3 are important:
ATK contributes the most to CP which ranks you higher in gym, HP makes your mon last longer which means more damage dealt and DEF is self explaining.

for an Attacker it depends on your preference, ATK shortens the fight, DEF is not very important but makes you use less potion and HP is just important if you have trouble cleaning the gym. Maybe I should mention if you prestige a friendly gym it is useful to have low CP mons to get more prestige, so it can be useful to have an attacker with low ATK

All IVs are useful. However different IVs are more useful than others depending on what you are going to have the pokemon do.

Defense:
HP IV is the most useful because HP gets doubled.
Att IV for gym placement, and punishing any attackers.
Def IV for just having your mon live a bit longer.

For Attackers:
Att IV for faster killing.
Def IV for better potion efficiency.
HP IV is not particularly useful.

For Prestigers it actually goes Def, HP, then Att because you want as many stats as you can get for as little CP gain as possible. Also good prestigers rely more on high DPS moves rather than attack stats.

All IVs have benefits, the benefits that each one provides depends on the base stats of your Pokemon.

Pokemon gain different benefits from different IVs, because it is the relative increase that is important - Pokemon with lower base stats gain more from the respective IV.

Stamina is guaranteed to be useful, because a Pokemon that has the same stats but with more HP will last longer before fainting, while attack and defence may not necessarily affect your damage dealt / received, due to rounding in damage.

It is possible for a Pokemon with 0 IVs to deal and take equal damage as a Pokemon of the same level but with higher attack/def IVs because of rounding.

Some Pokemon benefit much more from defence than stamina, such as those that have already high base stamina values, examples being Snorlax, Lapras, Chansey, and Wigglytuff, while other Pokemon benefit much more from stamina because of their low base stamina values, such as Alakazam, Magneton and Cloyster.

Defence is less useful on attackers due to dodging. If you're already reducing damage by 75%, damage values are smaller, defence is less likely to make an impact.

For example, lets say in a specific Pokemon matchup, the difference of 15 def IV reduces your damage taken by an extra 10%.

If you dodge, and you're normally taking 2.5 damage per quick move which gets rounded to 3, you're now taking 2.25 damage which still gets rounded to 3, so it doesn't do anything.

But if you don't dodge, you're taking 10 damage a hit, then reducing it by 10% would reduce the damage to 9, which would make a difference.

HP being doubled doesn't make stamina IV any more useful (or less) compared to the other IV on the same Pokemon, because what matters is the relative difference that I mentioned earlier. Pokemon with high stamina will still gain more survivability from gaining 15 more defence than 15 more stamina.

For example a Pokemon that has normally has 100 HP, and has an increase of 10 HP due to its stamina IV has a 10% increase in HP.

On defence, that HP becomes 200 and the extra HP due to stamina IV becomes 20, but the relative increase is still the same at 10%.