Emergency Operations
Authority and communication
14 CFR 91.3 makes the pilot in command directly responsible for the operation and final authority, and permits deviation from the rules in an in-flight emergency to the extent required. You do not need ATC permission to deviate - you declare the emergency and ATC supports you. Squawk 7700, use MAYDAY for distress and PAN-PAN for urgency, and 121.5 as the emergency frequency. ATC's job is to provide assistance; separation responsibility for VFR aircraft in an emergency is still the pilot's.
Engine failure
In cruise, the memory sequence is roughly: pitch for best glide, establish a field and turn toward it, attempt a restart, communicate, and prepare for the forced landing, with the checklist out only once the aircraft is under control. Best glide speed is published in the POH and gives maximum range, not maximum time. An engine failure on takeoff below a safe altitude is normally a landing straight ahead or within a shallow turn - not a turn back to the runway.
Systems failures
Pitot-static blockage symptoms differ by port: a blocked pitot with a clear static shows an airspeed indicator that behaves like an altimeter (climbs in a climb); a blocked static gives an airspeed indicator that under-reads in a climb and over-reads in a descent. Vacuum failure leaves attitude and heading indicators unreliable, so cover them and fly by the remaining instruments with partial-panel scan.
Fire, ditching, ELT
Know the POH memory items for engine fire in flight (fuel selector off, mixture idle cutoff, master off after the checklist says so) and for electrical fire (master off, then ventilate). Ditching: land parallel to the swell, into the wind if possible, on the downswell side. The ELT is tested only during the first 5 minutes of any hour.
Trap
Checklist paralysis. If the aircraft is not under control, fly it first - aviate, navigate, communicate, in that order.