Determine adequate speed of flyback diode for a relay
This has nothing to do with a relay, other than its coil acts like a inductor.  What you are really asking is how to chose the flyback diode across a inductor.

There are three main parameters to look at:

Voltage rating.  This is the maximum voltage the diode can take across it backwards, and still block current and not get damaged.  This must be at least the maximum voltage applied to the coil.

Current rating.  The maximum current thru the diode will be the same current that is going thru the coil when the coil drive is shut off abruptly.  The maximum coil current must already be known to design the coil driver.  Ideally the diode should be rated for at least this current.

However, many diodes allow significantly higher currents for short times than the maximum allowed continuous current.  This can be relevant in the case of a flyback diode.  Flyback current will decay on its own, so if the coil is shut off only occasionally, it can be valid to consider the pulse current spec instead of the continuous current spec.  If you are not sure how to calculate all this, use the continuous current rating.

Reverse recovery time.  This is how long it takes the diode to switch from conducting to non-conducting mode.  If forward current is going thru a diode and you instantaneously change the voltage so that the diode is reverse biased, the diode will actually conduct in the reverse direction for a little while before it shuts off.

Now think of when this situation occurs when driving a coil.  If the coil was recently turned off and the flyback current is still flowing thru the diode and the coil driver is switched on again, then there is a short from the power supply thru the diode thru the coil driver until the diode catches up and stops conducting.

If you are driving something slow like a relay, this probably doesn't matter since the time from off to on is always long enough that the flyback current has died down.  However, in something like a switching power supply or a solenoid or motor being controlled by PWM, the off to on time can be a small fraction of the flyback current decay time.  In that case, you have to consider this carefully.

Big fat power diodes meant to rectify line frequency (50 or 60 Hz) can often have substantial reverse recovery times.  Sometimes the datasheet doesn't list this spec at all, since if it matters, you shouldn't be using that diode.  Try finding the reverse recovery time of a 1N4004, for example.  I just checked the On Semi datasheet, and it's not mentioned.  It even calls these "standard recovery" diodes, which is marketing speak for "These diode are slow, so slow that we're too embarrassed to even tell you.  But instead of being up front and calling them "slow", we'll call them "standard" and then everything else we sell will be "fast" or "ultra-fast" or "super-fast" or "turbo" or whatever other terms our interns can dream up because we think you're dumb enough so that giving something a cutesy name will make you buy more of them.".

There are rectifier diodes where reverse recovery has been taken into account, sometimes with terms like "fast" or "ultra fast" in their names.  Don't use the names to guess speed, but at least the actual speed will be listed in their datasheets.  For small currents, you can use small signal diodes, like the 1N4148, that have reverse recovery time of only a few nanoseconds.  Schottky diodes are usually so fast as to be effectively instantaneous to most circuits.