Americans Hate Tingle: A number of American Gundam fans (the UC fans in particular) really, really loathe this series, declaring SEED Destiny to be the worst Gundam series ever made. Why? Because of this one particular character's Hostile Show Takeover (you know who he is.)
Americans are also annoyed with Kira's and Lacus' peace-or-else behavior turning them The Scrappy for a large number of fans.
Author Avatar: Kira & Lacus for Mitsuo Fukuda & Chiaki Morosawa, respectively. Morosawa was rumored to have cosplayed as Lacus at various anime conventions in Japan, and tried to get her husband to cosplay as Kira. Here's a fan-made pic of them to see what they would look like.
Author's Saving Throw: The four-part compilation movie 'Mobile Suit Gundam SEED Destiny: Special Edition' retelling the entire series from Athrun's perspective with some Revision. See the list of changes at The Other Wiki.
Broken Base: Do you even need to ask? Hell, even the staff didn't like how it turned out. Matt Hill and Sam Vincent -- the English dub actors for Kira and Athrun -- have expressed dissatisfaction with how the series was handled, while Kenichi Suzumura, the Japanese actor for Shinn, has stated that the Super Robot Wars Z incarnation of Shinn is "The True Shinn Asuka" as far as he's concerned.
Complete Monster: Lord Djibril's plans fail so often that he would have been nothing more than a Smug Snake if not for the fact that he successfully and enthusiastically mass murders hundreds of thousands of civilians with WMDs. Twice. He even pulled the trigger personally the second time and bragged about having guts to do this.
Compressed Adaptation: The manga. The 50-episode series gets condensed into 4 volumes (19 chapters), and becomes incomprehensible at times due to the heavily rushed pacing.
Critic Proof: Gundam SEED Destiny is very popular among the Japanese audience, despite the disappointing reviews the series received.
Crowning Music of Awesome: The debatable quality of the series does not extend to Toshihiko Sahashi's soundtrack, which is some of the best composed for anime with a seamless combination of orchestral, electronic, rock, and cloral numbers. The theme songs are no slouches either (with the arguable exception of the 4th opening).
The 4th opening's actually remarkably good, it's just not a song for an opening. The real problem with the 4th opening (not to say it worked as an opening), was that it came at the point in the series which is generally accepted to be where the show Jumped the Shark, which was also not helped by the fact that the new ending theme and previous opening themes were still much stronger than it was, so Wings of Words never stood a chance.
Hostile Show Takeover: Some profess that Kira becomes the de facto main character, limiting Shinn to a secondary role.
Internet Backdraft: Oh dear Lord, the flame wars started by the mere mention of this series, as well as Kira and Lacus' names in fan forums...
Jerkass Woobie: Shinn's an unpleasant little jerk a lot of the time, but that doesn't make what he went through any less horrific. You want to hug him and slap him at the same time. To a lesser degree, Auel and Sting seem to count for some fans since neither of them (plus their "sister" Stella) had the chance to get anything similar to a normal upbringing and that causes them lots of fatal trouble.
Moral Event Horizon: For some fans, Neo Roanoke crossed it when he put Stella into the Destroy Gundam, particularly as he'd previously promised Shinn to keep her out of combat. Other fans are more willing to give him a pass on account of the fact that he was brainwashed by Blue Cosmos.
Never Live It Down: During one of her first sorties with the Impulse, Luna has an opportunity to destroy a shuttle carrying a fleeing Djibril into space. She misses and he escapes. This one thing has eternally defined her character for some people, though it was already established in episode 7 that Luna was a lousy shot during her target practice sessions.
Gundam Seed Destiny itself is considered a huge case of Never Live It Down. After seven long years since the end of the series, and even after it's better handling in the Super Robot Wars games, haters will never get over their hate towards the series and its characters.
Nightmare Fuel: Djibril's vaporization, one of the most graphic of such deaths in the franchise.
Not to mention the episode where the Minerva crew finds the facility that made the Extended pilots like Sting, Stella, and Auel, which actually causes Arthur to vomit.
Replacement Scrappy: Shinn, who seems to have been an attempt at creating a darker, more conflicted Kira. It didn't work. Or more accurately, the production crew mistook being less popular than Kira for being outright hated, despite Shinn consistently ranking just behind the top stars of SEED in popularity polls (and ahead of fan favorite secondary characters like Yzak).
Ron the Death Eater: Fans who disagree with Kira and Lacus' tactics often forget their motives and try to turn Kira into Rau 2.0. The actual Rau 2.0, Rey, also suffers from this at the hands of fans who Flanderise him into a Complete Monster, while forgetting his genuine friendship with Shinn, admiration for Durandal, and good intentions.
Scapegoat Creator: The level of hatred that the most rabid anti-fans deliver towards Fukuda and Morosawa sometimes reaches disturbing levels. It doesn't help that they have a reputation as Lying Creators that may be entirely due to poor fan translations of the relevant interviews.
The Scrappy: The Broken Base may fight over just about every aspect of Destiny, but just about nobody on either side likes the DOM Trooper pilots. Poor characterization plus overuse of Zeon Expies equals Black Tri-Stars clones who spout lines like "You'd have to be pretty messed up to disagree with Lacus-sama!" Shinn fans hate them for being one-dimensional zealots, while Kira fans hate them for Not Helping Their Case.
Strangled by the Red String: Shinn's and Lunamaria's respective first loves are Stellar and Athrun. Stellar dies tragically, Athrun, along with Lunamaria's sister, Meyrin, defect, are shot down by Shinn and presumed to be dead. They got better. Lunamaria and Shinn end up together but due to the way it was handled, they are widely believed to have been this. The Special Edition is a little better about this.
They Wasted a Perfectly Good Plot: Basically nothing about Neo Roanoke's identity is played to its full potential; most of the implications (such as the parallels between his situation and the Destiny Plan) are never even addressed.
Alternate Character Interpretation: Yuna was always a coward, who was very skilled at manipulating a distraught eighteen year-old girl who was dependent on he and his uncle for political support, but failed the minute things didn't go his way.
What an Idiot!: While most instances can be easily disputed, Athrun has a severe case of Aesop Amnesia that causes him to make poor decisions in situations where fans think he should have known better. However, Athrun's Patriotic Fervor may have also been a factor in the amnesia's onset.