Stable Games
Well, I am still not sure of what you asking about by stable.
If you mean stability of the true count from hand to hand the 6 & 8 deck games are best. As the count changes slowly there (at least until you near the end of the shoe) one can hide his counting as you will increase your bet size slowly as well.
If you are refering to longevity of casinos offering a game, I would think the most commonplace game, 6 deck, will be around the longest. True, blackjack has changed, albeit slowly, over the years. In the 50's blackjack was single deck dealt face up to the last card. In the 60's single deck games began to be dealt face down to only 75% penetration, and Northern Nevada began to limit double downs to 10's & 11's. In the 80's the 4 deck game became prevalent. Now, it is 6 deck dealt to about 75%, often times with dealer hitting a soft 17, and the single deck game, when you can find it, is only dealt to 1/2 of the deck in downtown Vegas casinos. The evolution has been towards more decks and poorer penetration. At the same time, counters have come up with better methods to maintain their advantage.
But given the rate of change, I would think blackjack as we play it now will be around for several years to come, with the only changes being to the casino's shaving more edge for themselves. What you can do to stay ahead is improve your game by learning a stronger counting system, adding shuffletracking, or a few of the shady tricks some authors mention in their books.