Rare Replay

From the description it would seem like any Smash Hit would work, however it's either glitched or needs a better description as I have only found one way to make the counter in the Milestones screen go up.

First, you have to knock down a small enemy (throwing him will also work) and have the dizzy effect going on where birds fly over the enemy's head. When this happens, move over him, press down and mash the attack button repeatedly . Your 'Toad should start beating on the enemy if he has any amount of health left, then go to his smash hit attack (or go right to the smash hit attack if the enemy is low enough on health). Zitz will drill the enemy, Rash will claw the enemy's throat out, and Pimple's foot will become a weight that he crushes the enemy with.

This doesn't seem to work with large enemies as the 'Toads will boot them off screen. It's also hard to do it with weak enemies as they typically die in one hit before becoming dizzy.

The fastest way to grind this would be a spot with a few rats in level two (there are a few such areas) where you can easily take them out since there are only a few on the screen here at a time. What you can do is save the game, beat the rats in your area of choice, then reload the save and beat them over and over again.

I'm going to elaborate from what iamtfc said as of now only the downward smashes work whether this will be changed who knows.

The fastest way is using level 2 but instead of reloading a group of enemies over and over get the enemy count down to one and make this rat dizzy.

Save the game once you are stood next to the dizzy rat the nearer you are the easier it is. Then keep reloading that save 100 times if you miss the attack use the rewind feature it's a grind but it's the most effective way

The fastest way is to make a save at the start of level 3 (space ship interior) then jump & strike (B then A) at a rat enemy which will knock him down & make him dizzy (stars spinning round his head), quickly run up to him & repeatedly press the attack button (A).
If done correctly your chosen battle toad will perform his special finishing smash attack : Rash (throat rip) Zitz (drill to the face) & pimple (T ton wieght foot).
Keep doing this on the rat enemies upto the point of reaching the first appearing ladder then reload the save to speed things up.
Using this method i was averaging 15 smash attacks per run every few mins.

Note : the purple rats seem to be too weak to perform smash attacks on.

The fastest way I've found is to save right before the final hit of the smash move and load over and over after the enemy turns to bones.

To do a smash you just need to find a small enemy preferably a rat with more health and nock it over it should have stars above its head, now just walk up to it and press + and you should start the smash attack it will ether punch or do the final hit immediately depending on how much health is left in the enemy.
If it punches just keep hitting until the final move starts.