The following is all pertinent to the 777 as we operate them at Emirates. There is no requirement to check synoptics at any time neither for normal nor non-normal operations. The plane will alert you via an EICAS message for any non normal situation that requires your attention and direct you to the proper checklist (in most cases anyway). Status messages are not something you're required to do anything about as such other than to consider dispatch implications once you get to your destinations.
As for normal operations we're required to do time and fuel checks at least once every 60 minutes but most people I've flown with normally uses +-30 minutes. Other than that monitor radio and flight path (routing, airspace, terrain, weather etc) - and for myself I always play the game "What would I do if XYZ happened right now - what is my plan B and C"