Starting with Android 5 Lollipop (on my Nexus 7), I can't silence notifications from the Stack Exchange app without turning off sound completely, which also disables alarms.
If I understand this answer on the Android site correctly, the problem is that the app uses the "media" channel rather than the "notification" channel for notifications.
The specific problem I'm having is that I have notifications from the app enabled, with the default sound. When I put the tablet into "Priority interruptions only" mode (which disables audible notifications for events, reminders, and messages), I still get audible notifications. Setting the tablet to "No interruptions" is not a solution, since I use it as an alarm clock.
As a workaround, I've set "Sound" to "Silent" in the app settings. This should avoid audible notifications while I'm trying to sleep, but I actually want audible notifications during the day.