This is what I've discovered: You don't get helicopter storage until Act 2 at the Saints Penthouse HQ, so it doesn't matter which ones you find in Act 1.
The Airport Crib is only for airplanes, not helicopters.
The helipads at the airport are raised structures on the western end.
The helipads at the National Guard armory have Eagles , in both armed and unarmed models, if you do try to steal one, make sure you run off with the armed one (you can see the missile pods on the sides).
Helicopter Homie will deliver a Tornado to you, put that in your helipad at Saint's HQ to keep it instead of having to call him all the time.
This was the first combat copter I managed to keep (I did this before finishing the "Trojan Whores" mission).
"Trojan Whores", as @FallenAngelEyes said, will unlock a Vulture .
Finishing all the "Heli Assault" diversions will not unlock a helicopter.
STAG has two VTOL aircraft, both are stored at your helipads.
One is a VTOL fighter with a laser cannon, the other is a troop transport.
gaming.stackexchange.com/users/9300/CyberSkull
The earliest I have found that you can unlock a helicopter is during the main storyline.
Once you complete the mission Trojan Whores , you receive a Vulture helicopter for the mission reward.
It's accessible at the Saints HQ pad in Sunset Park.
Just go outside next to the pool (after you're inside the HQ, so the outside of the penthouse) and you'll see the floating key indicating a garage, which is where you'll be able to store future helicopters.
I was level 15 when I acquired this, which is 9 levels earlier than you can receive the Heli Homie.
I'd also done a bunch of activities on the side to raise my level though, so if you just go straight for the Trojan Whores mission, you'll unlock it earlier.
gaming.stackexchange.com/users/160/FAE
You can get several helicopters anytime, but you need to have access to the Saints HQ Penthouse to store them.
Outside the penthouse, past the pool, there is a helipad.
Take the helicopter there, then once you land on it, press E (or whichever key) to store it.
There are two different helicopters at the airport, along the west building (along the water).
On the army base island, in the middle yard with 4 helipads, there are 7 Eagles and 1 Eagle with miniguns and rockets.
The weapon variant Eagle is on the southeast helipad.
Also on the army base island, on the helipad on the roof of the building, there is a Tornado (so you can just store that one, instead of buying heli delivery).
Once STAG arrives in Steelport, you can get a few VTOL aircraft.
Althought they fly like a plane (and an alternate hover setting)
you can still store and keep them in the Saints HQ penthouse helipad.
gaming.stackexchange.com/users/17372/AsSeenOnTV
if you are looking to find the helicopters at the airport
they're the Stork (big cargo helicopter) and the Thompson (little ugly one)
but you have to be on foot or in a land vehicle for them to spawn.
If you go to the army base the eagles will spawn all the time but the Tornado (army paint) is on top of the building, I got it by skydiving onto the top of the building.
If you use cheats get the RC possessor and the money cheat and upgrade it to level 4 then you can control any helicopter.
If you want the red (Morningstar) tornado you need to get maximum wanted level on Morningstar then it will come after you but you need to use the RC possessor
gaming.stackexchange.com/users/20864/tylerfernando
If you go to the very northwestern part of the map, you will find a huge ship on some sand and not in the water.
On the very back of that boat, there is the Thompson helicopter.
It's not very good, and it's pretty ugly, but it's still a helicopter for personal use.
I found this and stole two police helicopters BEFORE doing the Trojan Whores mission.
gaming.stackexchange.com/users/17037/zach
The earliest you can fly a helicopter is at the after the third mission (We're Going to Need Guns) when you get Shaundi's Ex's Apartment.
Go to the airport (North-East corner of the map) and go to the North-West corner of the runways.
Here you will find two helipads.
The North one will have a Thompson, and the South one will have a Stork.
The earliest you can get a helipad for storage is after the fifth mission (Party Time) when you get the Saint's Penthouse HQ.
It won't come with any helicopters, you'll have to go to the airport and fly them back yourself.
gaming.stackexchange.com/users/18965/System Folder
How do I get helicopters for my personal use outside of activities in  Saints Row: The Third ? I've bought the hangar, but I don't see any copters lying around the airport. Do I just need to look around more or complete a specific mission/activity, or level up and buy the Helicopter Homie?  And once I've found one, where do I garage them?
How do I unlock helicopters?
gaming.stackexchange.com
<saints-row-the-third>
gaming.stackexchange.com/questions/36746
gaming.stackexchange.com/users/9300/CyberSkull
You can find the Stork and Thompson helicopters at the airport. You need access to the Saint HQ Penthouse to store them. The earliest you can fly a helicopter is after the third mission. Helicopters could be acquired either by exploring certain locations or by completing certain missions.
A helicopter could be found in several locations such as the airport or certain helipads. It is suggested that a helicopter could be acquired by completing some missions.