A broken spoke on most wheels means that you're waiting for a ride. Heather, Mike, and Sam taking a little break at the industrial park.
Tires are the easiest and most critical things to check, but I've learned to be alert to phantom shifts, as that may mean you've got a cable breaking, one strand at a time. Mine have always failed at the shifter. You can shift to your biggest cog, then without turning the crank click down to what would be the smallest cog. That puts enough slack in the cable to allow you to pull back the housing at the shifter so that you can inspect the cable where it leaves the shifter. If you see any broken strands, replace the cables. If a rear derailleur cable fails on the road, you can pull the cable to a cog you can live with, then tie the cable off to the housing boss. A front der' cable breaking just means that you ride home in the small ring. New cables and housings can give your shifting that nice, crisp feel, so it's good policy to replace them at least every couple of seasons, depending on you level of use.
There seems to have been a small outbreak of broken spokes in my ride circle. I don't know how you can anticipate a broken spoke, but if my recent experience is broadly true, most low spoke-count wheels are unridable upon the failure of one spoke. I think it's due to a combination of fairly extreme wheel warp and the closely spaced chain stays of modern bikes. I always thought I'd be able to open the brake and ride home in the event of a broken spoke. I reconsidered that idea a few weeks ago as I trudged barefoot up JFK with my shoes in one hand and my bike slung over my shoulder with the other. My thanks to the passing rider who picked me up, BTW.