Flyback diode requirements
A slow diode like a 1N400x can be fast enough, but you haven't provided any specifics to decide that.  The issue of diode speed is how fast they turn off, not on.  A slow diode can be used as long as the inductor is not turned on next before the current has stopped and the diode therefore no longer conducting.

The 1N4148 is a fast diode, but can't handle much current.  Again, you haven't given any particulars to indicate whether it would be acceptable in whatever your situation is.

If the voltage is low enough, usually a Schottky is a better choice since it will have very fast recovery time.  Schottkys have more leakage though, especially at high temperature.  Again, there is no way to tell from your question what you actually need.