I'd love to add push support to my social reader. Getting notifications for webmentions would be really nice. Sites where you kick off some kind of job/process could notify you about changes in status, sites for delivery tracking could notify you about status updates.
I think I did it once and it started to annoy me after some days ^^
I was always wondering if anybody does use it effectively. For me there are mails, calendar & pc updates the only things I want my computer to notify me. My phone does the rest.
@aaronpk I keep my notifications from apps very minimal (mostly just chat mentions, and hide almost all app icon badges), so for browser notifications it would be similar... Web applications where something is kind of important to see soon.