An electronic device may have a flexible display such as an organic light-emitting diode display. A strain sensing resistor may be formed on a bent tail portion of the flexible display to gather strain measurements. Resistance measurement circuitry in a display driver integrated circuit may make resistance measurements on the strain sensing resistor and a temperature compensation resistor to measure strain. A crack detection line may be formed from an elongated pair of traces that are coupled at their ends to form a loop. The crack detection line may run along a peripheral edge of the flexible display. Crack detection circuitry may monitor the resistance of the crack detection line to detect cracks. The crack detection circuitry may include switches that adjust the length of the crack detection line and thereby allow resistances to be measured for different segments of the line.