I've noticed many topics on the matter, but they are all incomplete, and I wanted to make the ultimate one, where I try to make a list of all playable hidden planes in WT (that means, excluding Ju52, C47 and so forth).
The equivalent list for ground vehicles can be found here http://forum.warthunder.com/index.php?/topic/262140-hidden-playable-tanks-the-complete-list/
1. What is considered a "hidden" plane?
A hidden plane is a plane which can be found in the regular tree or in the premium section, but is visible only to those who already own it.
2. How does a plane become hidden?
Many planes that were originally part of the regular tech trees were removed because they were redundant (e.g. SM.79bis 1942), too bad even for a 1.0 BR (e.g. Po-2) or for other reasons (e.g. O2SU Kingfisher). Others may be added in-game as hidden premiums from the very beginning (e.g. MBR-2).
3. How do I know if a plane I own is now hidden?
I'm trying to keep this list updated, but some planes might become hidden with upcoming updates. If you want to check if one of your planes became hidden recently, just create a new account and look for said plane in the tech trees. If you don't see it, it became hidden for new players and those who don't already own it.
4. I've noticed one plane in game, yet I can't find it in the tech tree, nor is it included in this list, what should I do?
Just comment here (if possible with a screenshot of the place where the aircraft should be in the tech tree) and I'll add that plane to the list.
5. How do I get one of those planes?
Most of them are now obtainable through events and challenges, or by buying them on the online store. They might become researchable (e.g. La-174) or purchasable with in-game premium currency (e.g. german Sherman tank) just for a couple of days. Just keep an eye on the news section of the forum. Some of them however are now history and will never come back: they will remain only in the accounts of those who started playing early and got to research and buy them.
Germany / Italy
He 51 C-1/Late 
He 51 B-2/H Hydroplane 
He 112 B-2/U2 (Romania) 
Bf 109 E-7/U2 
SM.79 1937 
Fw 189 A-1 
Bf 109 G-2 (Romania)
Hs 129 B-2 (Romania)
Me 262 A2-a
Great Britain / France / Australia
Gladiator Mk.IIS 
Boston Mk.I 
Plagis' Spitfire LF Mk IX
P-39N-0 (Pokryshkin) 
Spitfire Mk.IXc 
SB 2M-100A (Spain)
SB 2M-103 MV-3
Yer-2 M-105 TAT-BT
Yer-2 M-105R TAT-BT
OS2U-1 Kingfisher 
OS2U-3 Kingfisher 
Rasmussen's P-36A Hawk
XF5F Skyrocket 
Richard Bong's P-38J-15 
P-51 D-10 
F7F-3 Tigercat 
Ki-10-II C 
Ki-21-I hei 
HERE is a spreadsheet with the complete tech trees of War Thunder.
If you have any more planes that you know of, please write them below and I will add them. I'll try to keep this updated in case some of the above become visibile again or more hidden planes are added.