In D&D 3.5e, in my experience, multiclassing was a very common choice, and the prestige class system, where you're basically always going off of the base classes at high level anyway made taking a few levels of another base class seem like a relatively normal option — at every level, you're choosing which class to advance in, and hey, why not add a level of rogue? (Assuming favored class matched up and all that.)
Overall, I think it's generally the case that multiclassing was awful for spellcasters, but pretty nifty for martial characters — no real downsides, and a lot of customization. Ranger/Rogue split for the first 10 levels or so makes a nice "scout" time of character — arguably better than the "Scout" base class published later. In many ways, this felt like it meant that a dozen base classes equaled a thousand class possibilities (even if most of those were poor).
In making a D&D 5e character (starting at level 3 for a one-off play session), I was trying to look for interesting combinations, and kind of coming up blank. 5e softens the spellcaster problem (with shared "caster levels" and slots when you multiclass) and removes the alignment restrictions and favored classes in favor of (fairly low) ability prereqs. But, on the other side, 1) multiclassing now only grants a limited number of proficiencies for additional classes, and 2) there are increasingly-better class abilities as you go up, and 3) ability score boosts and feats are now tied to class levels — multiclassing means delaying (and forgoing) these.
So: are there 5e classes which do synergize nicely for multiclassing? Or do these all trade shiny, shiny versatility for being distinctly behind the curve? Is multiclassing a trap? (Still a thousand options, but now all of them poor?) Note that I'm not looking at loss of 20th-level capstone abilities, but low- and mid-level play (level 3 in the example at hand, but let's say also 5th and 10th.)