Notification settings live in Preferences > General, under Notifications:

• Show notifications for new unread messages — a desktop notification for each new message.
• Show notifications for repeated opens / clicks — whether read-receipt and link-tracking events notify you again after the first time (see repeated tracking events).
• Play sound when receiving new mail.
• Resurface messages to the top of the inbox when unsnoozing — see Snooze.
• Show badge on the app icon — Hide Badge, Show Unread Count, or Show Total Count, plus Count unread messages in all accounts, not just the selected folder.

Only your inbox triggers notifications

Mailspring notifies you about new messages that arrive in an Inbox, for every connected account. Messages that a mail rule or a server-side filter files straight into another folder or label don't produce a notification. If you want both the label and the notification, have your rule Apply Label without also archiving or moving the message. To at least see that those folders have new mail, turn on Show unread counts for all folders / labels in Preferences > General.

The tray / menu bar icon

Show icon in menu bar / system tray (Preferences > General) keeps Mailspring running and checking mail after you close the window. In Preferences > Appearance you can choose how the tray icon signals new mail (Blue icon for new and unread messages, Red icon for new and blue icon for unread messages, or No unread status indication) and, on Linux, whether it's drawn for a light or dark tray. On Linux the tray icon needs libappindicator or libayatana-appindicator installed, and you must restart Mailspring for the icon to disappear after turning it off.