Everybody's style is different, but for me, I silence... everything. Nothing dings. Nothing pops up. Not on my macbook, or my phone (Do Not Disturb, always), or even my microwave.

For Slack, I watch for a number in the red dot above its icon. Then I know I need to check it. For e-mail, sometimes e-mails just get by me. It's not usually a problem. For calendar, I have to keep a mental note of my next meeting. That is probably the biggest downside of this setup.