Strawman Has a Point: Kainan says the justification for wiping out the Moorwen was to tell themselves they were just animals, thereby implying that they weren't. But for pretty much the entire movie that's exactly what the Moorwen acts like and no more... a cunning but completely vicious animal. At best, the two are Not So Different, as the Moorwen retaliates for the death of its species by slaughtering those who had nothing to do with it, including people on an entirely different planet.
Geneva Dunsay: Spoilt girl who was used to getting what she wanted, and damn anyone who suffers for it? Or a spirited young woman trapped by circumstances and only wanting her first time to be special?
Frank. Part of the fandom thinks he's a victim of tragic circumstances and sympathise completely with him; another part hates him and considers him selfish and a Jerkass for everything he did to Claire during their twenty-year marriage. And then there are the fans who don't feel either way about him and those who like him but acknowledge his flaws.
Claire herself. Most people like her for her qualities, but some find her tendency to act without thinking annoying (especially when she does something which Jamie warns her beforehand not to do).
Jamie, mostly because of the (in)famous strapping scene. Some fans condemn him for beating Claire, while others accept his reasons for it and like him for his personality and his actions/decisions in other book scenes.
Ensemble Dark Horse: Murtagh, one-hundred percent. Fans were devastated when he was revealed to have died at Culloden. Among some of the TV-show watchers on social media, the hashtag "save Murtagh" is being circulated, and quite a few fans seem to be hoping for him to survive Culloden in the STARZ adaptation.
Lord John Grey proved to be so popular that he gained his own spin-off series, the Lord John books.
Germans Love David Hasselhoff: The books are very popular in Germany, for some reason, to the point of new books being published there translated into German on the same day as the English originals.
Hilarious in Hindsight: The proper Gaelic pronunciation of Laoghaire's name is something like "Lahiri." By the time the TV show made this clear, the name likely brought to mind Mindy Lahiri.
Love to Hate: Jack Randall, naturally, by most of the fandom.
Moral Event Horizon: In backstory, Jack Randall crosses this when he is revealed to have tortured and raped Alex MacGregor, causing MacGregor to suicide. During the main plot, readers first see just how twisted he is in the Wentworth scene, when he rapes and tortures Jamie Fraser.
Stephen Bonnet crosses it when he robs Jamie, Claire and their party of everything they own just hours after Jamie saved him from being recaptured and returned to prison.
Nightmare Fuel: Contains more than a little, no surprise. High points include the lovely medical scenes.
Official Couple: Jamie/Claire, Brianna/Roger, Jenny/Ian, Fergus/Marsali, Frank/Claire in the beginning of Outlander.
Then as of An Echo in the Bone there's also Rachel Hunter/(Young) Ian and Dottie/Denzell.
Brianna/Roger was teased a lot in Dragonfly in Amber, with Roger and Bree being instantly attracted to one another and Claire even assuming that they would eventually start a relationship together. Her main comfort in leaving Bree to go back to Jamie was that she knew that Roger would be there to look after her. They were eventually upgraded to Official Couple status come Drums of Autumn.
Voyager begins with Jamie being Not Quite Dead, which although it was stated at the end of Dragonfly in Amber it's still a shock to read as the very first scene of the book. He is forced to hide in an abandoned house with his comrades and later watches as they are taken out and shot by the English for being rebels. Jamie's own life is spared, because of a vow William Grey had made in Carryarick when Jamie spared his life, and so Jamie ends up being sent home to Lallybroch by William Grey's own older brother, Lord Melton. After that, Jamie spends seven years living in a cave while hiding from the English, before finally giving himself up to help his people make it through the famine. He goes to prison, but is later indentured in Helwater (a much nicer option than being sent to the New World). In Helwater, Jamie is blackmailed into sleeping with his master's spoilt daughter (who is getting married soon and doesn't want her first time to be with an elderly man like Ellesmere) and ends up accidentally getting her pregnant and fathers a son, and he is forced to kill Geneva's husband Ellesmere to prevent the baby from being thrown out a window (Geneva's husband was not a very nice person, especially after he discovered what she had done with Jamie). Jamie, much like the reader, is considerably shocked by it all. Much, much later, when Claire finally comes back to him, we find out that Jamie married Laoghaire, the woman who purposefully plotted to have Claire arrested and sentenced to death by fire as a witch (though in his defense he would never have agreed to wed her if he had known that) and Claire understandably freaks out and runs away and while she is away Laoghaire shoots Jamie which causes Claire to come back. While they are reconciling and trying to fix up the Laoghaire mess Young Ian is kidnapped by pirates while diving for the seal's gold (which Jamie hoped to use to recompense Laoghaire) and the rest of the novel is Jamie and Claire racing to find and rescue him.
The Woobie: Jamie, full stop. Every book introduces some new torment that he has to go through. In the first three books alone, he is raped by Black Jack Randall, accused of treason and murder and is hunted because of it, almost dies from fever and serious injuries, and is almost killed at Culloden and find himself separated from Claire and is completely miserable for twenty years.
Roger, to an unimaginable degree. The 18th century must really hate him— from the moment he steps through the stones he is mistaken as a rapist, beaten to a pulp and sold as a slave to Indians, and later hanged and almost killed mistakenly just because he chastely kissed another (very vindictive) man's wife.
Brianna in Drums of Autumn, though YMMV. Her mother abandoned her to go back to her one true love in the eighteenth century, leaving her an orphan. She'd also been told that she's not who she thought she was, and that Frank, who she adored, was not her biological father, thus causing her inner conflict between accepting her real heritage or denying it (as accepting Jamie as her father would feel like a betrayal to Frank). Then once she finally decides to travel back in time to the 1700s to warn Jamie and Claire about the death-notice and finally chooses to embrace her Scottish heritage she ends up raped and pregnant and unable to return to her own time like she originally planned.
Actor Shipping: Due to their famous chemistry, a lot people wanted Sam Heughan and Caitriona Balfe to become a couple, and a few even became upset that they were revealed to be just friends and dating other people.
Base-Breaking Character: Claire's character in the TV adaptation has become subject to this, as one section of fans complain about her attitude and her personality, which is portrayed differently from that of Book Claire. It doesn't help that she's 'mothering' Jamie and telling him what to do half the time in the STARZ series, either. However, there are other fans who accept Claire as she is in the TV series and don't pass judgement on her behaviour.
Broken Base: Some book fans were quite upset that Dougal makes an unambiguous Attempted Rape on Claire that she has to knock him out to get away from, followed by the entire aftermath of the equivalent and far less aggressive scene in the book being kept exactly the same. Especially since there's quite an overlap with the fans of Game of Thrones, which at the same time was also having issues of turning up the rape content of its source material.
Ensemble Dark Horse: Le Comte St. Germain, the Duke of Sandringham, Mother Hildegarde, Mary Hawkins and Louise de Rohan.
Informed Attractiveness: Mrs. Fitzgibbons gushes over Claire, saying that she hasn't seen skin so clear and unblemished on anyone else over eight years old. Yes, Claire has beautiful skin, but we can see that nearly every other character has equally clear and unblemished skin.
Les Yay: Geillis and Claire have a bit of this, certainly evident during their witch trial, when both Claire and Geillis are willing to die for one another. Claire even has a way out if she just claims Geillis corrupted her, but she can't bring herself to do it. Geillis then saves Claire by confessing to witchcraft and corrupting Claire. While the angry mob carries her away Jamie is able to run away with Claire.
Mexicans Love Speedy Gonzales: Averted. While the show isn't disliked in Scotland, it is far less popular there than it is in other places (particularly North America). This is in large part because despite the fact that it is set in Scotland, most Scots still consider it to be an American show, and it isn't nearly as large a part of the modern Scottish cultural consciousness as shows such as Still Game, Limmy's Show, or Burnistoun.
Name's the Same: Geillis Duncan, a Scotswoman, shares her name with a real Scottish historical figure who was also accused of witchcraft. Her confession under torture in 1590 naming other women as accomplices helped usher in British witch trials.
Unintentionally Unsympathetic: A fairly large body of TV fans considered Claire to be this when she returns to Frank. This is in large part because Frank is given a major case of Adaptational Heroism in the show, while in the books he demonstrates sexist and racist views.
The Woobie: As of season 2? Frank. He has lost the wife he deeply loves for two year, refusing to believe that she left him for another man. When she gets back, she is firmly out of love with him, treats him coldly and Frank has to learn she's been another man's wife (and is pregnant by him). As a result, he stumbles out of the room and destroys everything in another room in clear grief and emotional anguish. It's very, very hard not to feel bad for the guy whose only crime seems to be not being a hunky Scotsman from the past.
Some Fridge Logic here: If Claire is cold against Frank upon her return, it is because she has a number of bad experiences with his strikingly similar forfather. To return to the affection of a spitting image of the man who has tried to rape her and worse, might be hard for her. Not that this is easy to explain, though.