Short version
There is no public- or semi-public way to know when the Ring/Silent switch is flipped, nor to query its current position.
Apple has designed the audio system so that you can (and should) let the OS do the muting for you, but you cannot examine or be notified of the switch itself. Any solution that does so depends on private framework symbols and will be rejected by App Review.

Detailed explanation

1. Public API
   • AVAudioSession, Core Audio, UIKit and the other public frameworks expose no property, notification, callback, KVO-able key or route change that reflects the Ring/Silent switch.
   • The only change you can observe is the behaviour of your own audio: if you use the categories “Ambient” or “Solo Ambient” the sound simply does not reach the speaker while the switch is in Silent.
   • Apple’s own documentation (QA1417 “Detecting the mute switch state on iPhone”) states explicitly:
   “Your application cannot programmatically determine the state of the Ring/Silent switch.”

2. Private API
   • Classes such as AVSystemController, SpringBoardServices, GraphicsServices, etc. do have symbols that mention ringer or mute state, but they belong to private frameworks.
   • Any use of them requires you to link against private frameworks or call undocumented selectors obtained with NSClassFromString / performSelector:. Either way your binary will be rejected by automatic scanning or by human review.
   • Shipping an app that relies on those APIs is therefore not viable for the App Store.

3. Polling tricks
   Before iOS 5 developers used a hack that looked at the old AudioSession “route” string, or played a 0-length/very‐quiet sound and timed the callback. All of these stopped working years ago, are unreliable, waste CPU/battery and still give you no instant notification. Apple discourages them.

4. “But application X does it”
   Applications that show their own “muted” badge either
   • never went through App Review with that code (e.g. internal/enterprise builds),
   • shipped before the current automatic checks and haven’t been updated since, or
   • simply guess (for example: show the badge when volume == 0 and hide it otherwise).
   None of those strategies give you a supported way to detect the hardware switch.

5. Recommended approach
   • Keep using AVAudioSessionCategorySoloAmbient (or, on iOS 10+, AVAudioSessionCategoryAmbient with the “mixWithOthers” option) so audio is automatically suppressed.
   • Do not add your own UI for the hardware switch—you cannot guarantee it will be correct, and Apple’s HIG asks you not to duplicate hardware indicators.
   • If you want to warn the user that sound is inaudible, the only reliable indicator you can use is the system output volume (obtainable via MPVolumeView or — for read-only — AVAudioSession.outputVolume). This is orthogonal to the silent switch but at least it is public and legal.

Conclusion
Requirement 1 (respect the switch) is solved by the audio-session category you are already using.
Requirement 2 (show an icon) cannot be implemented with any API that Apple allows in App Store software.