What’s the highest iOS iPhone 4s can run?

It does Support Apple music, and the absolute highest version supported is iOS 9.3.5. It's highly recommended you not update to it though, as the OS is quite slow on 9.3.5. Good luck!

"According to the ever-invaluable MacTracker:

Original OS: iOS 5.0 (9A334)Maximum OS: iOS 9.3.5 (13G36)

According to an Apple Community Specialist on Apple's Support Forum, as of February 2016 you could use Apple Music with 9.2.1, so you should still be able to access at least some of the Apple Music feature set with 9.3.5, at least for now.

As with any server-side resource, there's always the risk that the service provider will change the service in the future in ways that require newer versions of the client-side player apps than you can run. Much of Apple's website is no longer viewable in Safari unless you have OSX 10.10/10.11 or iOS 9; the same process has happened with iTunes streaming media. I'd assume that eventually Apple will change something at their end which your older player won't recognize; with luck, that won't happen until your old phone dies for good.

You might also want to be selective about which resources you use, given the 802.11n WiFi/3G cellular speeds of the device. High-bandwidth video might stutter a bit, but audio should work well. CD-quality (AKA Red Book) audio is 7.7MB/second, while 802.11n is 6.75MB/second; most streaming audio is compressed to well below Red Book level, so you should be fine."

Last was iOS 9.3.6, released July 22, 2019

You can use iOS version 8.4.1 as it is faster than iOS 9 in terms of overall performance.