Few examples of how I use their Heavy Melee.
Banshee:
- When she's Warp Balling turn it on for when the visual connects - variable with lag - then shut it off and get back to doing damage.
- When she's doing her AoE Scream and you know you're going to get hit tap it on for the initial burst. It breaks your bubble but soaks some damage.
- Don't try to use it when she's in her claw swipe animation. That you just really need to soak the shieldboost and gtfo. The reason is that she can become continuously looped in that animation, and it can kill you by endless stagger.
Ravager
- Great for soaking swarmers on your windshield when you're out of rockets.
Dragoon
- Will stagger him in his wind up smash animation. Will stagger his two friends in their smash animations too.
1v3 Dragoon Hit Squad: Best Used With A Shotgun
When you see them...
Shieldboost (You're going to take damage.)
Run at them in a completely comical fashion as the lone volus charges headlong into their midst. Try to gather them as evenly as possible.
Tap Heavy Melee on then off.
All three get punted back.
Choose the ugliest one.
Aim at it.
Shoot where you're going to hit (center mass, Headshot, whatever works)
His friends are probably back on their feet again.
Position yourself between them.
Repeat.
...This is one of those circumstances where heavy melee isn't used to soak damage, but as Crowd Control. You have to soak the damage with Shieldboost, which normally isn't the greatest way to do things. But if it lets you get up close to the Goonsquad where you can singlehandedly start tossing them around and making shortrange shotgun kills it's worth it. It's messy as tactics go though. It's definitely not the mid-long range RHA standard.
Atlas
- Incoming rocket, Shields up!
Prime
- Drone incin stopper.
If you ever take a DoT due to AoE damage you can Heavy Melee to counteract the damage if you're close and gated, or just have the time to kill.
Essentially, for most of the enemy units in the game it's a spammable aoe stun that has no diminishing returns and costs nothing to use except a short amount of time. You can literally stunlock the enemy. You have to play him at close range though.
Thanks for the useful answers. My lasting memory of using heavy melee was when my lancer ran out of ammo and a Cerberus Trooper was in front of me readying the melee baton. I throw up heavy melee and I notice that I cannot move nor get out of the situation. So, it seems to me that most volus kits is about staying away and out of harms way while dealing the most amount of damage possible. Obviously, I don't know how to use the heavy melee, but if the enemy finds a way to be standing right in front of you, then, it seems like you'd be playing the volus wrong, no?
Close range volus is great. 
***
I don't take the Volus Protector, aka...

...very often. The only time I really enjoy playing him is hosting, which I rarely do.
When I do play him, I spec for power damage, and bonus charge on Biotic Charge. I attempt to play an eye-level game of pin-ball.
***
Shock and Detonation Drone/Decoy can do some hefty damage output but are probably the hardest things to get damage done with. Decoy works extremely well on some enemies and is almost completely useless against other enemies.
For my part, the enemies it works on (Primes, Phantoms) give it enough of a strategic edge to keep it in the arsenal.
***
I hate the Batglove on all my Volus now. But it's definitely something to play at least once in your life. (Albeit maybe not on Platinum.)