At what current and voltage do typical resistors explode?
In my experience, resistors and diodes burn, but they don't explode.

The only components that I've experienced exploding are tantalum capacitors when placed with the wrong polarity and transient voltage suppressors ("tranzorbs") when exposed to ~2x their rated voltage.

To determine which resistors or diodes to use, you can check their power rating. For example a typical through-hole resistor is rated for 0.25 W. Depending on how you're using the resistor, there are three equations that are useful to calculate the power dissipation:


Watts = volts * amps
Watts = volts * volts / ohms
Watts = amps * amps * ohms


For example, suppose you have a 10k resistor that you're using as a pull-up resistor for a pin on a 5 V microprocessor. The maximum power drawn by the chip through the resistor (by equation #2) will be 5 * 5 / 10000 = 0.0025 W, or 2.5 mW. That's fine even the tiniest surface mount resistors.