NOTE: For some reason, not all hidden objectives are marked '(Secret)'. I've listed objective names verbatim (including typos), and included any objective marked '(Secret)' or one that is not required to complete the level and could be missed in a normal playthrough.

These were all found by reading compressed game files, and (usually) corroborated through play.

Roads to Moscow: In the very back of the map, there is a lone officer with his car and a couple entrenched infantry. You can either eliminate him with tanks, or lob shells from the SU-100Y until you receive the objective 'German Officer (Secret)'.
Organized Defense: There is a broken railroad bridge on your left flank. Some German engineers are trying to repair it. Eliminate them for the objective 'The Bridge Repairs Interrupted (Secret)'.
The Klin Ledge: There are some abandoned howitzers south of your initial position, but inaccessible due to the destroyed bridge. If you make it the long way around and capture them, you'll complete the objective 'Long-Ranged Artillery Captured (Secret)'.

Sapun Mountain: There are some unmanned howitzers in plain view in the bottom corner of the map. Capture them (or let the AI spawn some crews for you) to get the objective 'Abandoned Gun (Secret)'.
Battery Defense: There are some supply trucks in the left corner of the map, near the isolated houses with crates on the ground. Move near them to capture them to get the objective 'Supply Trucks'.
Cover of Darkness: There are a pair of Panzer IIs in the field immediately north of your position. Killing one will enable the objective 'Destroy the Enemy Vehicles', and killing the other will complete it.
Sebastopol Defense:
Secret #1: There is a heavy mortar southwest of the point with the rocket trucks. Either push tanks past the treeline to kill it or launch rockets in that location until you get the objective 'Enemy Monster Weapon Destroyed (Secret)'.
Secret #2: There is a rail line that goes from the German spawn point to an unoccupied peninsula. At the end of the line on that peninsula is an idle armored train. Recover it for 'Armored Train Salvaged (Secret)'.

Steel Avalanche: turns out you can use the engineering trucks you get at the beginning of the level to repair the broken railroad bridges *very* slowly. On that isolated island, you will find an armored train, getting the objective 'Armored Train (Secret)'.
Covert Action:
Secret #1: Your mission description mentions a group already lost. Somewhere northwest of your current position is an isolated barn. Approaching the barn will trigger a scripted execution scene. Disrupt the execution to rescue the Elite squads and get the objective 'Lost Group (Secret)'.
Secret #2: North of Secret #1 is a couple campfires. The northernmost campfire has an unmanned Elefant tank destroyer. Eliminate the infantry around it to capture, and get the objective 'Captured Tanks (Secret)'.
Bottle Stopper: Capture the German key point to get the objective 'Counter Attack', and a free armored train for the trouble.

Spring Cleaning: There is a hill with a couple tents between your two northernmost points. It hosts some artillery. Eliminate it for the objective 'Destroy Hostile Artillery (Secret)'.
The Strongest Link: Take both German key points to get the objective 'The Raid (Secret)', which draws enemy units away from the headquarters.
Dodgeball: There is a tent and radar array northwest of your starting position. Clear the area of enemies to save the captured American officers, get the objective 'Confusion In The Air' and stop the 'confused' allied bombing raids.
Pigeon Shoot: Kill every unit in every wave for the objective 'Total Destruction (Secret)'.

In Talon's Reach: Take the fuel depot in the northwest of the map for free paratroopers and the objective 'Pleasant Surprise (Secret)'.
Forced Eviction: A couple warehouses to the east host unoccupied mortars. Capture them for the objective 'Mortar Storage (Secret)'.

Desert Barracuda: Not any objectives associated with it, but there is a tank crew that can be saved in the bottom corner, POWs to be released near the radio station, and a sniper to be saved further along.
Smokeout: After clearing the roadblock, take the town. If you're quick enough, there will be a British officer you can capture. Garrison him in the lighthouse next to the command vehicle to trick the British into bombing themselves and get the objective 'Jam Enemy Communications'.
Precious Prelude: Destroy a bunch of trucks near the northern warehouse for 'Enemy Supplies (Secret)'. The AI is a bit weird in this map, and I suspect some might try to leave and despawn, preventing collection of the objective. There's a warehouse to the north, some howitzers to the west, unoccupied mortars in the center and a British point that when captured gives you some Italian tanks. Oddly enough, going to the southeast corner (that looks oddly snowy) right next to the point will give you a KV-2, which is invulnerable to everything on the map except artillery and grenades. A nice easter egg. Credit to BloodSurg for helping to verify the secret trigger as well as the KV-2.
Homecoming: There are some wrecks west of the lake, with some unoccupied British artillery. Capture it for the objective 'Salvaging The Wreckage (Secret)'.

Chapter 3
Big Brother: There are some unoccupied Heavy AA southeast of the Tigers. Capture it for the objective 'Russian AAAs (Secret)'.
Precious Cargo: There's a damaged Tiger in the bottom of the map. Repair it to recover it and get the objective 'Healing A Tiger (Secret)'.
Angry Nursemaid: About halfway through the trail, in the center of the map, there are some bunkers covered by Soviet infantry. Clear the infantry to free your allied squads and get the objective 'Hidden Power (Secret)'.
Volga Factories: Once the last of your reinforcements spawn, the Soviets will spawn 4 Katyusha rocket trucks at the munitions factory, and they will try to make their way to back lines. Eliminating all of them while they are en transit gives the objective "Cutting The Firepower (Secret)".

Survival: There's supposed to be an objective to capture the radar for 'Capture The Radar (Secret)', but it did not trigger in testing. The description also states that it's supposed to unlock V2s, which start unlocked, so possibly a leftover from a previous iteration.
Street Sweeper: There's a castle in the top of the map, next to the artillery. Eliminate all the enemies for the objective 'Secure The Castle (Secret)'.
Gatekeeper: Cross the railroad bridge towards the enemy point, following the railroad. Approach a train to capture it. Bring it safely back to your lines for the objective 'Capture Supplies (Secret)', and receive some heavy armor.
Heavy Hand: There is a US point in the middle of the map. Capture it for 'Capture The Monastery' and get some MLRS for free.

East Wind, Rain: Recover some uncrewed weapons to the right of the beach for the objective 'Discovered Weapon (Secret)'.
The Lengthening Shadow: Simply walk up to the officer right at the start and he'll let you borrow a couple Stuarts, and give the objective 'A Friendly Chat (Secret)'.
Fish in a Barrel: A plane crashes right as the level starts. Save the pilot and you'll get a free bombing raid and the objective 'Air Crush Survivor (Secret)'.

Diversion: Bottom corner of the map has an unoccupied Japanese tank. Clear the nearby infantry to capture it and get the objective 'The Captured Japanese Tank (Secret)'
Cornered Rat: Simply save 10 or more trucks for the objective 'Save Ten Or More Trucks (Secret)'. Unfortunately, there's only 7 to start, so good luck with that.
Chokepoint: Henderson Field: There's supposed to be an objective 'Cover the Allies (Secret)' that should give free Ground Attack, and there is a jeep containing an officer that drives to your keypoint and despawns, but I've never seen this triggered.
The Eagle's Talons: Kill every tank in both columns, get the objective 'Destroy All The Tanks (Secret)'.

Tooth and Nail: Capture some of the artillery pieces after 'eleminating' the crews for the objective 'Capture Artillery (Secret)'.
Battle of Leyte: Take the warehouses on the south of the island and clear the enemy for the objective 'Interrupt The Evacuation (Secret)'.

Chapter 4:
The King of the Hill: Put the three officers on the hill into the car and get them to the safety of the main base. They'll give you free artillery and the objective 'Saved Officers (Secret)'.
Showdown at the Bridge: Capturing points on this map will give free American units. Capturing the one directly south of the starting area will give those units as well as British ones, in addition to the objective 'Clearing the bottleneck (Secret)'.
Airborne Fury: East of the town is a weakly guarded warehouse. Capture the engineers for 'Captured Engeneers (Secret)'.
Lost and Found: Capture the fuel depot for 'Fire And Flames (Secret)'.
A Needle in a Haystack: There are German squads patrolling the areas where the downed officer might be. Killing this wandering squad will give the objective 'Delayed Problems (Secret)'.
Rail War: This allegedly has an objective 'Secret Objective - AA Battery' that would trigger B-29s, but killing the AA battery or the trucks with Flak 88s does nothing.
Battle of the Ruhr Pocket: This one is easy to miss. Your British allies are attacking in the north. Most of the time, you will get an objective called 'Destroy Encircled Enemy Troops', and the description will say something along the lines of your unskillful conduct means the British won't be able to help in the final assault. However, if you assist the British enough and they take few enough losses, you will have an objective of the same name, but it will instead praise you for your skillful conduct and give you 3 Centurions, their only appearance in the campaign.

That's all I've been able to find as text references in the game files. Let me know if any of these aren't working, or work differently than described. If there's interest, I'll try rooting around in the files for the expansions as well.