Red Sophia gives up fighting by the time she marries Cerebus.
Cerebus can be seen to go through this as well, but where it happens is debatable.
Bile Fascination: Many check out the later issues that contain the creator's legendarily misogynistic rants just to see if they're really that bad. They are.
Broken Base: Many fans of early Cerebus disliked the direction it took with High Society and Church & State. Many fans of High Society and Church & State disliked the direction Cerebus took afterward. There's also the debate about whether Dave Sim's anti-feminist views are misogynistic, misunderstood, accurate, or irrelevant to the comic.
Darkness-Induced Audience Apathy: The Church & State arc. The few sympathetic characters present (Michelle, Storm'send, The Regency Elf, etc) are quickly phased out; everyone who sticks around is either too amoral or too apathetic to really care about.
Designated Hero: Cerebus can be seen as this, since nearly everything he does is for his own selfish gain.
Designated Monkey: Dave Sim did this a lot as Cerebus went on, tying into his Creator Breakdown. Due to his rising misogyny, he hated pretty much all his female characters and generally wrote them out to replace them with Straw Feminist shrews. This, however, is nothing compared to Cerebus himself. A good part of Minds is devoted to Dave forcibly making Cerebus realize what an asshole he is and how totally unfit he is for human company. Pretty much the entire comic from that point on (about 100 issues or so, depending on where you think this begins) details Cerebus' slow and gradual self-destruction.
Growing the Beard: Happens a few times. Firstly, after Dave tripped out on LSD and created the first "Mind Game" issue where the comic began to have an actual plot. Also when Gerhard came in to do the backgrounds and inking, allowing Dave to concentrate more on what he liked doing and increasing the quality of the artwork dramatically.
Iron Woobie: The real Cirin has been through a lot. She was betrayed by her best friend; her peaceful movement morphed into a violent, fascist mockery of what it used to be; her mouth was sewn shut (which scarred her lips); she was placed under permanent house arrest; yet she bears it all with a smile.
Cerebus, after the events of Church & State. He pretty much stays this way for the remainder of the comic. Despite all the horrible things he's done, you can't help but feel at least a little sorry for him.
Jaka throughout Jaka's Story. She has an abortion because she thought a baby would make her ugly, and treats both her husband and Cerebus very badly but in the end your heart still breaks for her.
Pope Cerebus killed an infant and an old person to shock the crowds.
Cerebus raping Astoria was this for many readers, although her comment about it in Women hints that she may have deliberately goaded Cerebus into doing it to try to get pregnant by him and become even more of a thorn in Cirin's side.
Jaka's abortion is clearly intended to be this for Rick in-story, and possibly for the reader, especially in light of her telling Cerebus in an earlier issue that she feared having a baby would make her ugly.
Serna betraying her friend Cirin and stealing her identity.
Padding: In one issue, 4 pages are devoted to Cerebus taking a piss. Dave Sim seems to find this sort of thing incredibly amusing.
Protection from Editors: Cerebus was self-published. The closest thing Sim had to an editor was Deni, and when she quit, well, let's just say that no sane editor (and not many of the insane ones either) would have allowed Sim to do the things he ultimately ended up doing with the comic.
A few readers disliked the direction the series took after the first 25 issues (Volume 1), back when Cerebus was still a relatively unambitious parody of Conan the Barbarian. Not counting the controversial politics the series became infamous for much later on, some were turned off by the more complex narratives of High Society on onward, which generally had very little action and typical fantasy genre storytelling.
Not all readers were pleased with the narrative after High Society (Volume 2). While they appreciated the heavier political themes and more mature writing of those issues, some disliked the increasingly "cosmic" nature that the series was taking, especially the ending of Church and State. Not to mention just how unlikable as a character Cerebus had become throughout those volumes, his vile actions including throwing a baby onto the street, a crippled old man off a building, and perhaps worst of all raping Astoria in the dungeon.
Those that still appreciated the series during Church and State were later dissuaded by the following volumes, Jaka's Story and Melmoth. In the former, Cerebus was reduced to a side character, while the side character Jaka was given full focus, while the latter wasn't about Cerebus or the rest of the series at all, but rather a fictional representation of the death of Oscar Wilde.
And after Jaka's Story and Melmoth, the most common part of the series that spurns readers is the highly controversial Mothers and Daughters arc. At first, readers turned away from the series by the previous arcs that focused on unrelated side plots were excited at the prospect of an action-packed story returning Cerebus to his main character role. But they were bitterly disappointed as the whole arc became a polemic for Sim's controversial anti-feminist views.
Finally, the consensus of perhaps most readers is that everything up until Minds is at least tolerable, and essentially enjoyable. How far into the series one goes is largely dependent on either their degree of sympathy for Dave's politics, or their ability to ignore them as they show up in the work.
On the topic of what sends readers away from the series, we should also look at the only aspect of the series to be more or less universally praised throughout: Gerhard's beautiful background illustrations.
Uncanny Valley: Sheshep Ankh. He looks perfectly human, until you notice that his feet only have three toes. Also, in the author's notes for the phonebook of The Last Day, Sim noted that he purposely played around with some of the lighting and furniture in Cerebus' bedroom during his talk with Sheshep to give the hint to anyone that noticed that there was something very wrong about Sheshep.
Unintentionally Unsympathetic: Rick in Jaka's Story. While there's no denying that Jaka would be a poor wife for anyone, he remains a Lazy Bum who can't be bothered to look for work most days. His conversations with Oscar all but state that not only did he only marry for her to get the son he has always wanted, but any daughters she might have would be unloved and neglected. His eventually becoming an Author Avatar for Sim's gender politics only further poisons the well.
Red Sophia tearfully talks to Cerebus about their marital problems, not in the least bit noticing that Cerebus is in some kind of magical trance and has his head in a glowing sphere.
Jaka hears the reasons why Cerebus is being shunned by his hometown (namely, not being there for his father's death and funeral because he was running around with a woman), and then hits Cerebus with what amounts to "it's your fault for not doing what you wanted to do." He doesn't take it well at all.
What Do You Mean, It's Not for Kids?: Don't let the funny animal on most of the covers fool you. The first two phonebooks might be appropriate for teens, but gory violence, nudity, graphic sexual material, and other frankly "adult" themes run throughout the rest of the story arcs, not to mention the more controversial material.