Android only: The default Gmail client can sync individual labels, but only notifies users about new inbox messages. A tiny little app, Gmail Label Notifier, offers customised notifications of new label messages.
If all this app did was bring in notifications for label messages, it'd be just good enough to survive the next Gmail app update. But the Label Notifier takes its notification job seriously. You can set whether new label messages vibrate, make a sound, flash an LED or trackball colour, and select colours for each label. For very serious labels (Oh, let's say, "Adam"), you can have your phone repeat the notification if not answered after some time, vibrate in a Morse-like pattern, and even override your phone's silent mode. For messages you really want to see, it's a go-to app.
Gmail Label Notifier cost 0.99 UK pounds. Like all Android apps, you get 24 hours to try it out with a return policy.
A Label Notifier for Android [Google Sites]