-After the initial waves of popcorn enemies, a pair of large airships with red eyeballs show up. Quickly killing them triggers a large wave of popcorn enemies randomly appearing from the top of the screen, along with the appearance of three gundam looking robots that pop up near the end of this bonus wave; note that the gundam robots hold shields which are considered destructible parts.

-Just before the boss, three fighter jets appear, followed by a trio of stealth bombers at the top of the screen, followed by a pair of stealth bombers at the bottom of the screen. Destroying the three fighter jets as soon as they appear on screen triggers an alt. wave where the stealth bombers appear spread out on completely different edges of the screen. The one time I triggered this I wasn't able to kill the stealth bombers and they left, leaving the screen empty for a noticeable period of time before the boss showed up. I would assume that if you can quickly kill the spread out stealth bombers you would trigger another wave of enemies for a nice boost to your score, but that's just speculation.

-If you have a 12x multiplier when the boss fight occurs, you'll face off against a much tougher red variant of the normal boss, which has denser, faster attack patterns and a few new attacks where it strafes along the top of the screen firing off green oblong bullets. (Triggering the red variant boss for the first time also gives you an achievement)

-After the tank at the start of the level there's a highway with three tanks on it initially. Just as the highway begins to scroll off-screen a red sportscar will quickly zoom on it from right to left. The car is destructable, and you'll earn an achievement for destroying it, but I'm not sure what the point value is, and it's very risky to try and destroy it since you have to pay attention to the enemies appearing from the top of the screen.

-If you destroy every destructable part on the giant battleship, your ship will quickly accelerate forward to a bonus fight against a slightly more aggressive version of the giant battleship you just destroyed. This battleship however scrolls off screen much quicker regardless of what you do.

*According to a tip from Grey Dragon (much appreciated!), if you can manage to destroy every part on this second battleship, you'll actually trigger a THIRD battleship, this time with even faster scrolling!

-If you destroy every destructable part on the giant battleship while holding a 12x multiplier your ship will accelerate forward to a bonus wave where medium sized fighter planes zoom in one at a time from the left and right alternatively (i.e: one ship appears from the left side, once you destroy it another copy appears from the right side, etc.). Be careful, they will fire at you, and you CAN crash into them. Several of these planes appear before the level ends.

*Based on recent playthroughs, it seems that once the final part of the battleship scrolls onto the screen, you must quickly finish destroying everything or else you won't trigger either secret event.

One of these bonus waves gives you an achievement, and I would assume the second wave with the fighter planes is more lucrative for score.

-After the second stream of popcorn enemies, tanks will start zooming in from the top and bottom corners of the screen. Destroying every tank that appears triggers an alt. route where you move to a dense forest which is suddenly destroyed to reveal a massive fleet of tanks that shoot at you as they move down off screen. This is followed by a fairly weak miniboss.
You also get an achievement for meeting this miniboss.

Edit:
The second stream of popcorn enemies seems to contain a hidden trigger for another alt route where the zooming tanks and other enemies quickly pour in from the bottom of the screen, followed by a narrow row of tank pairs. The trigger seems to involve using normal shot on the second popcorn stream...

-Near the end of the stage, there's a pair of large turret cannons:
There's a medium sized ordinary looking plane on the right side of the screen, just in front of the platforms holding up the aforementioned turret cannons.
Once it scrolls into your view it'll immediately begin to move underneath the turret cannon platforms, but you can destroy it before it does so.
Destroying the plane is worth a base value of 10,000 points (not sure if your multiplier affects this)
You also get an achievement for destroying this plane.