For a time in the Fire Emblem: Mystery of the Emblem fandom, people greatly preferred Marth with Catria instead of his canon girlfriend and eventual wife Caeda/Shiida, whom they preferred to pair with Ogma. In an aversion of Die for Our Ship, they get along with Marth/Caeda fans, though.
Palla's a bit of an oddball. For a while, the Fan-Preferred Couple for her was to get her with Abel, due to her unrequited crush and the common view that Abel was better off with her than Est (especially since they meet Uncertain Doom together). With the release of the remakes, though, Palla/Minerva has taken off and distinctly surpassed it, due in part to distaste for All Love Is Unrequited and in part their rather strong supports in New Mystery.
Fire Emblem: Genealogy of the Holy War has the Seliph/Julia pairing. Because Seliph and Julia are actually half-siblings, they can't be paired up through normal means, but rather start at a nearly full relationship meter that gradually decreases. Despite this, many fans prefer to pair them together as there are many Ship Tease moments between the two, mostly in the form of Shipper on Deck comments. Also, it is technically possible to circumvent the decreasing love meter by exploiting a glitch in the "jealousy" system.
Fire Emblem: The Binding Blade's two most popular ships, aside from the basically-canon Roy/Lilina, are Melady/Guinivere and Perceval/Elffin. Both Melady and Perceval are heavily devoted to their noble, and many of their supports deal with their relationship and their Hurting Hero nature, and Perceval adds the dynamic that his noble is in disguise and has to play around it. The characters are also extremely popular in their own right due to their great power and good designs. Melady cannot support Guinivere (as Guinivere can't be deployed in the campaign) and has a boyfriend, but as said boyfriend dies, it doesn't much impact the ship's popularity.
In Fire Emblem: The Blazing Blade, Lyn/Florina is probably each other's most popular pairing, considering that Florina Does Not Like Men initially and seems to be crushing hard on Lyn. It doesn't hurt that they have a paired ending if they A-rank each other's support, and said ending basically has them retire to live together.
It's not uncommon among fans of to note that Eirika/L'Arachel's supports have a stronger romantic arc than the canonically possible Ephiram/L'Arachel. Unsurprisingly, it's one of her most popular options, if not her most.
Ephiram can end up with a lot of women, but for the most part, the fandom seems to have fallen behind shipping him with the tragically doomed Lyon.
Fire Emblem Awakening, due to the massive amount of in-game marriage potential, generally doesn't apply, but there are some exceptions, mainly Basilio/Flavia, Maribelle/Lissa and Severa/Noire, both of whom have a very solid relationship in their Supports and in-story scenes, yet cannot be paired up.
At least in the west, for a time Inigo/Gerome was the preferred pairing for him on the yaoi side due to their dynamic together.
Lucina/Severa is one of the most popular yuri pairings despite the fact that the two cannot even support during the game.
Fire Emblem Fates has Ryoma and Scarlet, who receive a fair amount of Ship Tease, especially during Revelation, in which Scarlet ends up suffering a Plotline Death shortly after her introduction, and Scarlet can only support with the Avatar. Despite this, fans also enjoy pairing the two together.
In the JP community, Dimitri/Male Byleth is by far the most popular ship for male Byleth despite the fact that the two cannot actually S-support in the game.
Dimitri/Claude is one of the biggest examples of this in player base. The pair is incredibly popular, perhaps more so than any other same-sex pairing for Claude, despite the fact that the two cannot support each other due to the mechanics of the game. Its helped by the fact that the two are mostly on friendly terms during the game and that in the Drama CD that has been released, Claude also comments on Dimitris body and is noted by Dimitri to have been staring.
Despite Sylvain's skirt-chasing antics, his most popular pairing is with Felix, and vice versa for Felix, despite the latter's comparatively larger number of A-supports.note Most of Sylvains supports end at B, while Felix can have A-supports with almost every character he can support with The fact that their supports and paired ending have a heavy dosing of Ho Yay make it popular to assume even outside of a certain demographic that they both end up together. It is one of the biggest ships in the fanbase and as a testament to the ships popularity, it currently has the most fics on Ao3, beating out every other pairing in the game including ones involving the lords/the main character.
The most common ship for Ferdinand is with Hubert (and vice versa), due to their A+ support reading almost like an S support, romantic music and blushing included.
Hilda and Marianne are more commonly shipped together more than any of Hilda's available romance options due to Hilda's desire to take care of Marianne when she's struggling being adorable. They easily have the second-most fanart of any yuri pairing next to Edelgard and female Byleth.
Annette and Mercedes are commonly shipped together due to their Romantic Two-Girl Friendship status in game and the fact that their ending implies that they settle down together. Their A support in JPN reading like an S-support definitely adds fuel to the fire.
Dorothea has the most potential romantic endings besides Byleth, but her most popular is with Petra, with the pairing having a ton of fans.
While both don't have an A-Rank support in-game, fans like to pair Ingrid with Dorothea due to the dynamics they share in their supports and Ingrid giving Dorothea a ring at the end of their paralogue adds fuel to the fire. People who pair them tend to ignore the fact that Ingrid flat out rejects Dorotheas advances in the game, though if people who pair them acknowledge it, they depict Ingrid being Tsundere toward Dorothea.
There is a rather fervent fanbase for Dimitri/Edelgard (for those who aren't interested in pairing them with Byleth) despite that it is impossible to happen in-game due to the fact that one (or both) of them will always die no matter what route is chosen. The potential comes from the Star-Crossed Lovers vibes, plus the fact that they were childhood friends for a time. Edelgard may have severe memory problems that forced her to forget their shared past, but she has vague residual memories of having loved a young nobleman in Faergus, and as for Dimitri, he does remember her, and considers the time he spent with her to have been the happiest period of his life.