Since Aquanox 2 never tells you what the bonus objectives are unless you complete them, I went searching for a walkthrough online that would list them and their rewards. It turns out that the only near-complete source is a set of YouTube videos by flaminflamin, showing completion of all bonus objectives on each level that has them.
(There are also German and English walkthroughs out there, but their information is incomplete and they completely omit the mission rewards.)

Unfortunately, all of flaminflamin's videos have been pulled from YouTube. BUT! Yahoo's cache has preserved the pages, allowing me to grab the descriptions. Combining this with a tad of exploration, I have built this ever-so-lovely list.

A note on the numbers: there are 33 missions (two of which depend on the completion of specific bonus objectives much earlier in the storyline), and these numbers are mostly from flaminflamin's playthrough. I did things in a different order (and I believe flaminflamin left some optional missions out), but these numbers are a rough guide for when these missions show up.

(If I decide not to be lazy, I'll update the list with the normal rewards for other missions; this would primarily be useful in noting where to get the Hydra.)

EDIT: raymccullie below found a very handy list with both all of the rewards and tips on how to get them, created 3 months after my original post. It can be found here:
http://www.gamefaqs.com/pc/561418-aquanox-2-revelation/faqs/62085
I've left the original info below.

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Objectives & Rewards:
2 - Terror at Lima II
Mission Targets
NAV Point 1 cleared: Vendetta Sniper
NAV Point 2 cleared: Buzzer (2)
NAV Point 3 cleared: Ceramic Armor, Stanley (2)
Bonus Targets
NeoYuppie Bomber destroyed: TresherShark (3), Stanley (2)
Freighter secured: EMPactor

3 - Dr. Finch
Mission Target
Finch escorted: Plasma Gun, Stanley (2)
Bonus Targets
Crawler Unit destroyed: XT Plasma Gatling, TresherShark (3)
ELF-Node protected: 2000 Credits
All Crawlers destroyed: Hitman
Freighter paralyzed: Torpedo Magazine 1

4 - Stoney in a Fix
Mission Target
Stoney protected: EMPactor. FlashShark (2), Buzzer (3)
Bonus Target
Cingan destroyed: XT Plasma Gun

5 - Mighty Maggie
Mission Target
Ships paralyzed: Mighty Maggie, Vendetta, Hitman
Bonus Targets
Freighter located: BullShark (2)
Freighter paralyzed: FlashShark (6)

8 - The Race
Mission Target
Finish line reached: BullShark (4)
Bonus Target
Stoney defeated: XT Vendetta Gatling

9 - Machina Antarctica
Mission Target
Finch protected: Plasma Jet, FlashShark (2), Buzzer (4)
Bonus Target
Hank Bellows's life spared: XT Vendetta, (and a bonus mission later, in which you get the Hydra)

10 - Chow Lung
Mission Target
Harvester protected: EMPactor, Plasma Gun, BullShark (2), Buzzer (6)
Bonus Targets
Chow Lung destroyed: TigerShark (2)
Research station defended: Medium Block, 3000 Credits

11 - Swedenborg's Getaway
Mission Target
Swedenborg paralyzed: Stingray
Bonus Target
All pirates destroyed: Vendetta Gatling

12 - The Attack
Mission Targets
Harvester defended: Vendetta Sniper, BigBang 1 (2), Buzzer (4)
Enemy destroyed: Plasma Gun, TigerShark
Bonus Target
Freighter paralyzed: Small Block (2)

13 - The Convoy
Mission Targets
Met with convoy: Lacewing, 5000 Credits
Convoy protected: Vendetta Gatling
Bonus Targets
Pirate Scouts paralyzed: TresherShark (4)
Freighter secured: Doom Mortar

14 - Plunder
Mission Targets
Hangar secured: Buzzer (6)
Outpost cleared: TigerShark (4)
Stores emptied: Plasma Gatling
Bonus Targets
ATL-TechBomber paralyzed: TigerShark (4), FlasShark (4)
ELF-Node protected: 2500 Credits

15 - Asylum Patrol
Mission Target
Asylum defended: EMPactor, TigerShark (2), BullShark (2), Buzzer (4)
Bonus Targets
ELF-Node protected: 4200 Credits
Freighter paralyzed: TigerShark (6), BigBang 1 (3)

16 - Pirates
Mission Target
Asylum defended: 5000 Credits
Bonus Targets
Redbeard's life spared: XT EMPactor, (and a bonus mission later)
Freighter paralyzed: BigBang 1 (3)
Freighter paralyzed: TigerShark (4)
All pirates destroyed: XT Stingray

17 - Freeman
Mission Target
Freeman Protected: 3500 Credits
Bonus Targets
Freighter located: Vendetta Gatling, Stanley (10), tresherShark (5)
Atlanter Located: XT Plasma Jet

20 - To be or not to be...
Mission Targets
Mercenary Scouts destroyed: Doom Mortar, Maneater (2), FlashShark (2)
Bombs stopped: Plasma Jet, Buzzer (4)
Bonus Target
Hamlet taken out first: Plasma Gatling, Gun Slot

21: Crawlers and Angels
Mission Targets
Energy supply destroyed: Plasma Gatling, Maneater (2)
Angelina saved: Big Block, FlashShark (2), Buzzer (6)
Bonus Targets
All Crawlers destroyed: XT Doom Mortar
Freighter paralyzed: TresherShark (6)

26: Deimos
Mission Targets
Defenses taken out: Deimos
Eerie defended: Laser Gatling
Eerie escorted: Maneater (6), Buzzer (6)
Bonus Target
Shantyman followed: Torpedo Magazine 2

27: Drake's Revenge
Mission Target
Redbeard killed: XT Laser Gatling
Bonus Targets
Redbeard's accomplices killed: Hammeahead (2)
Freighter paralyzed: Maneater (6)

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thanks! very useful post!

http://www.gamefaqs.com/pc/561418-aquanox-2-revelation/faqs/62085

The FAQ post doesn't just tell you what they are but also the where and how.

Ooh, very nice. I played the game 3 months too early, it seems!

Thanks for pointing this out!

mmmm necrothread....
now thats out of the way, is it just me, or are these bonus missions insanely difficult? I struggled on the first mission's bonuses and the second mission's bonuses I find almost impossible. I am just totally outclassed. Anyone know how to complete the second mission with bonus objectives? or know of a guide? or is it simply not worth doing them?

Hmmm, does anyone knows the requirements for unlocking the missions "Pirates" and "Drake's Revenge" listed in that GameFaqs guide? I know "The Race" is unlocked by killing Cingam and "Redbeard's El Dorado" by sparring Hank Bellow, but the first two missions I mentioned did not trigger in my playthrough of the game.