How do manufacturers determine LED characteristics?
Just about all LEDs can take at least 20 mA. This is almost certainly true of anything in a discrete leaded case, like T1-3/4 or T1. Some small SMD parts may be rated for less.

I would use a 5V supply with a 330 &Omega; resistor in series. You are very unlikely to have LEDs that can be damaged by this. This will put at most 10 mA through the LED if you get it in the forwards orientation, or 5 V in reverse accross it for the backwards orientation. Neither should hurt ordinary LEDs. 10 mA will be enough for even inefficient LEDs to light up visibly on the bench (assuming typical indoor conditions).

Rig up a jig to show the voltage across the LED. Since the power voltage and resistor are known, you can calculate the current from this too. This will give you one datapoint for voltage and current, which should be good enough for most purposes.

If you don't see the LED light up in either orientation, then keep in mind it could be IR instead of dead. You can use most digital cameras to determine this because they can see the IR light.