Many people think that Sharpshooter and Great Weapon Master feats are broken, because the -5/+10 trade-off is too powerful at low levels. Heavy Armor Mastery and Polearm Master are suspect as well. I've not seen any of these feats in play so I, personally, can neither confirm nor deny their brokenness. Most of the feats I have seen in play are, in my opinion, super weak-sauce and not as good as increasing an ability score by 2 points. I'd call out Sentinel, Magic Initiate, and Inspiring Leader as feats that aren't broken. The rest are pretty suspect, or else I haven't seen them in play enough to decide.
I have seen druid Circle of the Moon in play, and it is extremely powerful at low levels. However it starts getting better balanced somewhere around level 6, and by level 10 is a total non-issue. I don't think I'd call it broken unless you're deliberately planning a campaign that sticks to very low levels.
Going in the other direction, the ranger Beast Master is pretty weak. It is definitely not unplayably weak -- I was in a group with such a ranger up through level 9, and the character was able to keep up with the party. But, it was a noticeably worse option than, uh, pretty much any other character in the party. Even the Champion fighter was having a lot more impact. (As an aside, I think Champion fighter is balanced just fine, having seen several in play. It may be weaker than the Battlemaster but both are well within the bounds of what I would consider "not broken.")
These lists of broken things don't usually include the Tempest Domain cleric but holy crap that guy is potent if built right. The ability to inflict damage when hit is very powerful, but holy crap, the maximizing of damage is absurd. I DM'ed one of these up through 12th level, and while I don't think I'd outright ban this ability, I wouldn't consider it unreasonable for another DM to do so.
I've never seen a Wild Magic sorcerer in play but I've heard that its power level fluctuates wildly based on the DM and how often they force you to roll for wild magic. So maybe overpowered, maybe underpowered; some people would say that anything which relies so heavily on DM preference is inherently broken.
Skill-wise, I'm of the opinion that Perception is ridiculously overpowered, especially with Expertise and/or the Observant feat. Such a character basically finds everything automatically. Stealth is also incredibly potent, since most enemies have total crap for their own passive Perception. Conversely, I've seen relatively little use of skills like Insight, Medicine, Nature and Performance, and probably a couple of others that I'm forgetting because I just never see anybody roll them. I understand the role of these skills in the game but they are so underutilized I wonder if they could have been packaged in some other way (e.g. minor class abilities, background features, or parts of feats).
I feel strongly that Expertise is broken as hell at higher levels. Of everything I've mentioned here, this is the only rule I might actually house-rule to work differently. The problem is that it goes outside the bounds of bounded accuracy, leading to situations where one character automatically succeeds while another struggles or automatically fails. Such a divergent modifier leads to weird decision-making. Auto-success and auto-failure should come from special abilities or something, not baked into the modifier, so that the difference is obvious up-front. YMMV; a lot of people seem to really like having super-high modifiers.
The races in 5e are pretty balanced but half-elf seems really strong (+2 Cha, two floating +1s, and +2 skills) and dragonborn seems pretty weak. Depending on how you feel about feats, some people think variant human might be overpowered or that regular human is underpowered. I personally don't find any of the races to be outright broken.
There are some ostensibly broken multiclass combos, mostly involving warlock, sorcerer, and/or paladin, but it's unclear to me just how bad they are. You can do something really horrifically overpowered with Life cleric plus goodberry, but that's veering into one-trick-pony territory. I'm sure somebody has been keeping track of these multiclass combos and can tell you more; I tend not to worry about them because most people I play with don't build characters that way.
