What are the most common ways to fry an arduino?
Some I can think of that people have done...


Connecting more than 6 volts to the 5V pin. (The two ATmega chips used on most Arduino boards have an absolute max of 6V.)
Connecting more than VCC (typically 5 volts) to any of the I/O pins.
Not current limiting the I/O pins to less than 40mA.  (E.g. driving a motor directly or not using a current limiting resistor with a LED)
Connecting less than 5 volts to GND and GND to Vin or 5V (reversed power). Vin and Barrel jack do have a protection diode, but it isn't always enough.
Connecting a 5 volt supply to the 5V pin while connected to USB.
Changing connections while powered  -- never do this!
Not using flyback diodes with inductive components like motors, relays, and solenoids.


On the Uno and Mega2560 boards, which use an ATmega8u2 or 16u2 for the USB to Serial converter, it seems like whenever someone accidentally connects 12V to an I/O pin it gets blown. I don't fully understand why, but it seems like a trend.