My group plays only once a month at most, and all have busy lives, so it seems to requires to much bother for them to try multiclassing.
Also, as Dungeon Master, I have a generous rule whereby any new character starts at the same level as the party, so if a player wants to try another class, he just creates a new character. The only penalty is that he does not get any of the items which his former character had. Fortunately, 5E requires no items at all to keep up with the encounters.
I myself often multiclass when I play because I try to recreate characters from old systems using 5E rules. For example, I might build a warlord using a Fighter with a bit of Cleric for healing; or build a Friar (from Dark Age of Camelot) with a Cleric and a Monk or Fighter; or a Bene Gesserit using various classes.