Bard, Fighter, Warlock
I love the Monk class, lots of opportunities for players to create interesting characters but 5e really brought out the love for Fighters. Same goes for Warlocks. The Warlock was a neat idea in 3e that was never really implemented well. I liked the 4e Warlock, but the 5e took those innovations a few steps further. The Fighter is just a solid class. You can play it all of the way through or multi-class and create really interesting concept characters. Warlocks are great for multi-classing as well.
The 4e Bards are hard to beat but 5e might be doing it. The basic Bard subclasses from the PHB are awesome! I've had two really kewl Bards that I got to play in different campaigns and they both turned out great. Spell Secrets are so versatile. There are lots of spells from other classes that allow the Bard to become something unique. Taking a few Ranger spells can turn you into a Bow Bard. There's a couple of great Smite spells that would make for a really good Monster Hunter. One of my favorite Bards I played in a 5e campaign was an investigator, modeled after Jessica Jones. I took a lot of Divination spells, some Melee Cantrips, and Enhance Abilities. I actually decided to go for College of Lore to be better at investigating. I used one of my Spell Secrets to steal Tenser's Transformation so I could be a combat monster when I needed to. Mind you, there were a lot of options here. I could have went College of Swords or College of Valor. Swords was looking mighty temping. Instead, I went with Variant Human and Magic: Initiate feat to give me a Shield Spell right away and some melee Cantrips. So I had a lot of Cantrips and spell variety from the beginning. Sadly, we only played to 13th level. The DM had a baby and didn't start playing again until around her 3rd birthday, but hasn't been DMing.