Broken Base: Several Inhumans fans dislike how the Inhumans are becoming popular at the expense of the Mutants and want to have both species stand side-by-side. Others are just grateful that the property is actually getting major recognition at all from Marvel.
It's Popular, Now It Sucks!: More like "It's Being Forcibly Dragged Into The Mainstream, Now It Sucks". Inhumans have long been regarded as one of the Marvel Universe's hidden gems, but now that the characters are being given a greater presence in the setting, some of that appeal has diminished. A lot of it, of course, has a lot to do with how they were dragged into the mainstream, and at the perceived expense of Mutants at that. Many have noted that the Inhumans being a Suspiciously Similar Substitute for the X-Men is very tin-eared noting that it's overall set-up, i.e. a Hidden Elf Village with Crystal Spires and Togas has more in common with say, Asgard, Wakanda, or the Fourth World (to cite some of Jack Kirby's creations which scholars see Inhumans as containing a dry-run of), than a largely urban teenage-centric superhero group who live in contemporary society, and the overall eugenicist based self-exclusion from humanity has nothing in common with the X-Men.
Overshadowed by Controversy: Notice how many of the examples listed here have to do with how this franchise has been promoted extensivelyat the expense of the X-Men franchise? There are good reasons for that; they mostly have to do with the fact that the Mutants have almost always had a larger fanbase than Inhumans, and the rest come as a result of the perception that the executive in charge of these decisions (Ike Perlmutter) is a controversial figure to begin with.
Replacement Scrappy: While they were pretty popular before, and individual Inhumans are still quite popular (e.g. Kamala Khan, the current Ms. Marvel), this is what most of the fans saw them as after Marvel tried to push them as a replacement for Mutants.
As this article notes, there is something disturbing about Marvel having a group that they've long used as a stand-in for the LGBT community getting wiped out in favor of the Inhumans, who've long represented intolerance and separatism. It is perhaps unsurprising that mutants have since rebounded spectacularly with the destruction of the Terrigen Mists, while the Inhuman Royals have vanished into deep space to find a replacement for the now destroyed Terrigen Mists.
Joe Glass, an author of several comics featuring gay superheroes, wrote this article criticizing the attempts to fit the Inhumans into the same Applicability mold as mutants. He compares the similarities between the various experiments used to create Inhumans to the implication that you're turned gay by an outside force, which is a very disturbing idea since Mutants have been a longstanding allegory to homosexuality.
For the Comics:
Broken Base: The post-Infinity era, headed by Charles Soule. Either it's a new golden age for the franchise, bringing them into the mainstream and giving them the attention they deserve or a Dork Age that's removing much of what makes the Inhumans unique and making them the poor man's X-Men, complete with the overexposure.
The Inhumans' status as social outcasts is largely self-imposed, as they possessed both advanced technology andStock Superpowers well back into ancient times when ordinary humans would not have had the means to persecute a society that had either, much less both. They practice Eugenics and maintained a sub-race of deformed, genetically engineered slaves, the Alpha Primitives. Their internal culture is a mess of class divisions (a fact which Maximus has exploited more than once) despite their comparatively small population. Plus, the current "heroic" royal family overthrew the previous monarch because they did not want to give up the Slave Engine, a device intended to transform humanity into Alpha Primitives! At best, their "heroism" is analogous to the characters of A Song of Ice and Fire (or its adaptation Game of Thrones). Even their status as Mutant analogues is dubious, since the majority of their mutations are activated deliberately via Terrigenesis, and if they were to forego the desire for superhuman powers they could still be Badass Normal types like Karnak even without it! Thus any Body Horror or Power Incontinence actually are their faults because, unlike Mutants, they get to choose to play the Superpower Lottery. Though much of their "less-than-stellar" aspects have been retconed, destroyed with the crashing of Old Attilan, or dissolved as part of Character Development. Especially their Fantastic Caste System and abolition of Alpha Primitives.
The Inhumans vs X-Men running plot, particularly Cyclops' involvement in it - namely, he was painted as mutant Hitler for destroying one Terrigen cloud (which, lest it be forgotten, sterilised or killed all Mutants it came into contact with) and then standing up to Black Bolt, before being vaporised. While it transpired that he'd already been dead long before any of this took place, having died of M-Pox from the cloud, and everything that followed was Emma Frost manipulating events with her powers to give him a martyr's death as part of her progressive Sanity Slippage, very few people actually knew that, and Black Bolt still vaporised what he thought was Cyclops. Of course, considering Cyclops' popularity as a Well-Intentioned Extremist rebel, it could have been a meta move on Marvel's part.
During Inhumans vs. X-Men, even a number of Inhumans admitted that the X-Men have a lot of very real bones to pick with the Inhumans, and narrative sympathy is definitely weighted towards the X-Men.
With fans of the X-Men. Originally the two didn't mind each other, but after a massive marketing push by Marvel billing the Inhumans as "the next big thing!", the two seemingly can't stand each other. This stems from the Inhumans going from their own unique thing to being Suspiciously Similar Substitutes for the entire mutant race, which many feel is because of Marvel not owning the film rights to the X-Men.
Following Inhumans vs. X-Men, it's got better, thanks to Medusa finding out just why the mutants have been driven to a last ditch attack (the Terrigen cloud is about to dissipate and infect the entire planet, rendering it uninhabitable for mutants), and deciding that Terrigenesis, sacred as it is, isn't worth the death of even one mutant, let alone all of them, destroying the remaining cloud. With that, the rebounding of the X-Men in ResurrXion, and the Inhuman Royals heading back off into space to find a replacement for Terrigen, meaning that the two groups are no longer in competition, relations have cooled significantly.
For that matter, "classic" Inhumans fans and "modern" Inhumans fans, mostly coming from the older fans who think the new breed destroyed the concept in the name of pandering to a nonexistent fanbase, and the newer ones who think the old Inhumans comics were an embarrassing relic that nobody really cared about. This is only exemplified by the Death of the Inhumans storyline with the main controversy being how the push into mainstream had destroyed the brand.
Genius Bonus: "Terrigen" is likely taken from "teratogenesis", literally meaning "monster birth". This is a Fridge Brilliance if it's intentional.
A subdued version of this can be seen again with Crystal and Ronan the Accuser.
Even Janet Pym called out Crystal and Johnny's relationship when Crystal called out Janet for wanting to wed the mysterious new Yellow Jacket from an old Avengers comic.
Tearjerker: The romance between Gorgon's daughter and Reyno, an Alpha Primitive child. Since she's the daughter of one of the royal family, she can't really be seen with him, and Gorgon's forcing her to go through terrigenesis (such a great father), which is changing her physically and mentally each time. The two run away to the Fantastic Four for help... and Reed refuses to do anything for fear of upsetting the Inhumans. So poor Alecto is taken back to Attilan and made to go through with the process anyway. Some time after, the boy tries approaching Alecto, only for her to harshly yell at the primitive daring to speak to her... only for her to suddenly break out crying, because she just can't keep up the pretence, and she still loves him.
Unintentionally Unsympathetic: The push into the mainstream, meant to garner new fans, soured a lot of people either because it came at the expense of the X-Men and mutants at large by bringing them to the brink of extinction with near-universal hatred again or because of the general mayhem and chaos of a lot of people's lives being ruined from being mutated, some to death. A lot of it is directed at Black Bolt, for unilaterally causing it in the first place after the last time him blowing up a terrigen bomb nearly screwed over the universe, with some spilling over to the royal family for being rather self-possessed in addressing it. There's a prior storyline in the Mighty Avengers where a king is deposed, replaced, and erased from history while given a lot of sympathetic traits, such as respect for their slave race, and the reason he was deposed being his decision to disable/remove the Slave Engine, an Inhuman gene-altering weapon created to mutate regular humans into more Alpha Primitive slaves, something that the other Inhumans denounced him for. The fact that they're quietly retconning Black Bolt into being megalomaniacal and the royal family into being more noble is rather transparent.
Voodoo Shark: Dan Slott's run of the Mighty Avengers introduced the Unspoken, an inhuman monarch who apparently ruled between Black Bolt's reign and that of his parents. He was dethroned by the royal family when he disarmed the Inhumans of a hidden super weapon left by the Kree capable of mutating all of humanity into Alpha Primitives. This coupled with denying the existence of it already makes them deeply unsympathetic. Then, later writers try and do an Author's Saving Throw and retcon the Unspoken into being Evil All Along while they were trying to stop him. During this they end up in New York, where they meet a teenaged Spider-Man. A problem with this is the royal family were all teenagers during the coup of the Unspoken, and they still are for this story. The Fantastic Four were all active when Spider-Man started. This would mean that in the window of time between meeting Spider-Man and meeting the Fantastic Four they all aged enough for Gorgon to have two children, one of which is at least a teenager.