Officially Apple does not support iOS 10 on any iPhone model prior to the iPhone 5. In addition, the iPhone 4s hardware is just not capable of utilising many of the features of iOS 10.
This is not to say it wouldn't theoretically be possible to do so using non-standard means of installation, but the difficulties in doing so far outweigh any benefits of doing so.
Also, iOS 9.3.5 (13G36) is the last version of iOS that is officially supported on the iPhone 4s and many users complain that the phone struggles with this version of iOS. In fact, there was even a class action lawsuit against Apple claiming that iOS 9 should not have been installable on the iPhone 4s because the hardware struggles with it. One can only imagine how much more the hardware would struggle with iOS 10.
[EDIT - A final note]
In my opinion the risk of an attempted installation of iOS 10 on an iPhone 4s is too great. While it works for some, it doesn't work for most - and for a small percentage it even makes the phone totally unusable (with reports of the phone no longer switching on or, if it does, messaging no longer working and/or it's incapable of being charged afterwards).
Unfortunately, it isn't clear (at least to me) why it works for some and not others, and why it totally bricks the phone in some cases (although I suspect it may have something to do with what version of iPhone 4s is being used).