Can I install iOS 10 on iPhone 4S?
There are some videos telling how to install iOS 10 on iPhone 4s, for example How to get iOS 10 on iPhone 4/4s/5/5c/5s/6 and above. Though they contain the term iPhone 4s in the title, I am not sure whether it works, because that video doesn't clearly show installing iOS 10 on 4S device.
The software update in iPhone 4S doesn't show the iOS 10 update, but is there a way to install it?
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).
You can't officially install IOS 10 on an iPhone 4s or below, but you can do what the guy in the video did or by jailbreaking your phone. The problem is it will probably brick your phone or, if not, make it really slow. If you want to get IOS 10 cheaply, your best bet is to buy an iPhone 5s or 5, which aren't too expensive right now!