Base-Breaking Character: PJ, mainly for being a poor replacement for Pixy as Galm 2. Some bemoan his ineptitude as your wingman in these missions, whereas others find it refreshing that he's not an Ace Pilot like Cipher or Pixy, but an idealist up in the skies fighting for his beliefs in spite of his relative lack of skill and talent. As such, while many were shaking their fists at Pixy for gunning PJ down at the very end of the game, there are several players who instead said good riddance. Due to the game's story encouraging Alternative Character Interpretation from all angles, this response could've very well been intended by the developers.
Mission 10: Mayhem is essentially Zero's version of Comona, a massive, purely air-to-air skirmish with you nearly single handedly wiping out the Belkan resistance, including one of three Belkan ace squadrons. In certain versions, it also features cameos from Jack Bartlett and Wolfgang Buchner.
Deconstruction: In-universe. Of the Belkan War. Those who played Ace Combat 5 first will be told that the evil Belkans invaded other countries and the heroic Allies liberated them. Well that happened, but after Ustio was liberated, Thompson says it perfectly, "The true nature of the war begins from this point forward." Not all Belkans were evil, as evidenced by the interviews. And the Allied Forces invaded Belka in the first place to secure "resource interests." Come Mission 11, Allied bombers destroy an entire Belkan city, and the Belkans rush frantically to defend it... before deciding to burn it anyway just so the Allies can't say they did. To quote Thompson again, "Everyone is a hero and a villain. And no one knows who's the victim or who's the aggressor." Pixy didn't know either. So he defected.
Demonic Spiders: Some of the "mooks" have a habit of flying just so close to your plane that you'll have to drastically slow down just to avoid overshooting (flying past), or will fly a tight loop to keep throwing off your own attempts to generate angles for missile shots or gunfire. Particularly annoying when it's in a skirmish, as decelerating to keep them in your line of fire will leave you vulnerable to other enemies' attacks, and deadly when (not if) the enemy aces do it to you.
The enemy aces in general. They almost always show up at the end of a mission, which if you have been spending it shooting down enemy planes may mean you are running low on ammunition. They will almost always outnumber you (with the exception of the Gelb and Espada Squadrons) and are exceptionally coordinated. As mentioned above, they fly in tight loops and while you are chasing them they often have someone else chasing you at the same time. At the higher difficulties, staying on top of them requires pinpoint accuracy to avoid blowing all of your ammo on nothing but the sky and co-ordination with your wingman to keep them from getting the jump on you. You don't get the latter on the Ace of AcesBonus Level.
Epileptic Trees: A recurrent piece of fan speculation concludes that Cipher could be none other than Mobius 1, despite the fact that Mobius 1 was considered a rookie pilot throughout the Usean Continental War with no reputation until the destruction of Stonehenge, making the idea highly unlikely outside of Cipher erasing his identity during the roughly eight-year gap between games. The fact that the default paint scheme for the F-15C in 04 is very similar to the default/canon scheme in Zero has partially fueled this.
He's Just Hiding: By virtue of Never Found the Body and no official confirmation from Namco about the character being KIA, there's a contingent of fans who say this about Pixy's replacement, PJ. Despite the loss of radio contact after Pixy's Morgan wastes them by scoring a direct hit through the Falcon's canopy with the TLS and the grave marker shown during the epilogue, players never see their plane go up in flames when they bail out: there's only a line of smoke as they fall out of the airspace and the glass seen after impact has led some to believe they managed to safely eject. In fact, if flying a Su-47 or another aircraft with a large enough bubble cockpit, you can clearly see their plane stabilize after PJ is hit, as if it was merely disabled. Others also point to the fact that the enemy aces interviewed throughout the story, including Solo Wing himself, were shown to have survived similar or worse circumstances, especially if they were shot out of the sky by the player's own lasers (which Cipher really shouldn't have access to anyway). More details can be found here. The Perfect Guide Book ultimately cuts the debate by confirming PJ's death.
The Burst Missiles used by Pixy's ADFX-02 Morgan during the final mission creates a large area of effect blast that deals continuous damage capable of destroying your plane in one go, often forcing you to break off your pursuit in order to avoid it. What makes it worse is, due to the PS2's graphics, it's invisible to you when you're actually inside of its range. Fortunately, it loads slower than regular missiles and stops being used after you land six hits.