The problem I find in terms of dealing with Musharna is the overall lack of pokemon that can effectively deal with it. Mind you, I don't find Musharna that hard to deal with, but seeing the same old monotonous Mushy counters on every team can get pretty dull after a while, damn skunks.
Bug-, Dark- and Ghost-types do the most damage to Musharna. Let's look at that. First, Bug-types swarm NU, reaching nearly 2 dozens, but they are either too SR weak and/or don't hit hard to begin with. Since Mushy commonly boosts its Special Defense, the most effective Bug-types to deal with Mushy are Pinsir, Leavanny (I swear, Aasgier XD), and Armaldo without the aid of boosting, though they by no means shouldn't. Quiver Dance Bugs can also work, but needs SR off the field if they don't want to be overwhelmed.
Ghost-types are few and far between in NU, and the only viable ones are Golurk, Misdreavus and Drifblim. The more common Acroblimp usually runs no Ghost moves, so it has little chance of taking out Musharna unless it times its Destiny Bond just right. Both Misdreavus and Drifblim can engage in CM wars, though these are less common than their standard sets, NP Misdreavus even more so. Golurk is really the only one who can severely damage Mushy without setting up with Iron Fist Shadow Punch, but those who lack the move may also struggle with the psychic fetus, not to mention it doesn't like boosted Psychics.
That said, Dark-types are the most effective way to dispatch Musharna, as they are immune to Musharna's strongest moves in Psychic or Stored Power. NU's Dark-types consist of: Skuntank, Liepard, Murkrow, Sneasel, Shiftry, Cacturne, Zweilous, Scraggy, Mightyena. Hey, that's a lot of Dark-types, surely Musharna can be contained easily right? Usually, if Mushy doesn't run a coverage move. The problem usually lies in Mushy Calm Minding and then striking, or outright hit the switch-in. Let's assess which of the 8 can deal with this. Cacturne and Shiftry cannot finish off a weakened Mushy with Sucker Punch as it will fail against Heal Bell, forcing them to target it with weaker moves. They are also slaughtered by Signal Beam, switch-in or not. Sneasel fares slightly better if it can catch a +1/+1 Mushy with Punishment, but even this can fail to KO a healthy Musharna who can KO with Signal Beam. Mightyena is in the same boat, but trades Punishment for Moxie (and Sucker Punch). However, Mightyena isn't strong enough off the bat unless it has the Moxie boost....which it usually gets by KOing a Dark-weak pokemon...yeah, you can kinda see the problem here. Murkrow usually can't do a thing back to Musharna (and hopefully vice versa), while Scraggy usually boosts exclusively on Mushy, as everything else in the tier can either hit it hard, outspeed it before several boosts, or wall it, which isn't ideal if you're trying to deal with one pokemon. That leaves Skuntank, Liepard and Zweilous. Skuntank is your most reliable bet, with Taunt, Pursuit, Sucker Punch and usable bulk to keep Mushy in check. However, this 3-move combo still isn't fullproof. I have heard several cases where Musharnas have escaped or even defeated the Skunk (perhaps involving T-Wave, Barrier, Baton Pas, or Hidden Power Ground) due to the Skunk user being outplayed. I didn't categorise Liepard into the "frail Dark-types that get KOed by Signal Beam) because Encore can stop Musharna from hitting you, should you switch in on anything else. T-Wave Liepards are pretty helpless against Mushy though, with T-Wave even backfiring against Synchronize, and if Musharna switches in on you we still have a problem. Zweilous is the only NU poke capable of performing the nigh impossible task of OHKOing Musharna turn 1 with CB Crunch, but the below average defenses, speed and accuracy is a turn off for most players.
A notable trend is that all super-effective moves against Musharna do not surpass 80 base power (no NU poke has STAB Megahorn), and is partly what makes Musharna the dominant force it is today. Heal Bell also crushes many players' hopes of stalling Mushy with status (Trick can work, but there's no second chance if the Mushy player predicts that). Despite all this, I still don't think Mushy is broken, just pretty annoying to bring down. Musharna shouldn't be moving up (it deals with Sawk like no other). If anything, Mandibuzz or Scoilpede (or to an extent Primeape) should get down here and silence (y)our Musharna troubles for good.