I come from Maths background, but recently I encountered problem where a channel is described as "flat fading channel". I would like to make some conclusions based on that fact. I suspect that for radio engineers this is self explanatory, but not for me. Do we prefer a channel to be flat fading vs selective fading? Can we even make the distinction here? If no what are the pros and cons of each?

A flat fading channel is one where the entire signal is scaled by the fading, i.e. the signal is multiplied by a gain $$a(t)$$ which (usually) varies more slowly than the bandwidth of the channel.