First of all, discount all MS. This will lessen headache of this debate. And now I set the scenarios.

Archangel
Captain: Captain Maryu Ramius (veteran of the previous war, commanded the Archangel during the most difficult part of its career and against insurmountable odds, emerged from the previous war intact. She is a very understanding captain while at the same time being a no nonsense captain during battle. Her personality endears most of the crew to her, which explains why many of the old crew still remain. Kira also treats her like a big sister. Very charismatic but a little weak against relationships.

Armanents:
"Lohengrin" positron blaster cannon x 2 (Most Heave weaponry, capable of firing two at once)
"Gottfried Mk.71" 225cm dual high-energy beam cannon x 2 (Standard Anto-Warship weapon)
"Valiant Mk.8" 110cm linear cannon x 2 (High speed firing RAIL gun often used in GS against MS)
"Igelstellung" 75mm automatic multi-barrel CIWS x 16 (useless except for destroying missles)
Torpedo launcher x ? (Wombat anti Ship missles, Sledgehammer heavy missiles used against MS and ships)
Large missile launcher x 24
Surface-to-air missile launcher x 16 (Helldarts, mostly for interception of incoming missles and non-PS MS)
Anti-beam depth charge launcher x 8 (Like hell they need them, AA's armour is laminated and extremely laser resistant.

Weaknesses: Apparently so far, only speed. Nazca's could outrun them in GS. Minerva on the other hand is more speed oriented so possess an advantage in this field.

Minerva
Captain Talia Gladys: We know nothing about Talia's prior combat experience except that apparently, she did fight in the previous war, making her a veteran. Her exploits on the planet Earth shows that she is a competent commander who knows her ship but her strategist skills, like Maryu is not displayed often. Personality wise, she is a little drab and not quite as likeable as Maryu. Her chief weakness is her inability to administer military discipline to her crew, as demonstrated by Shinn Asuka who has blatantly disregarded her orders many times and never shows her the respect that is due to a captain.

Nevertheless, like Maryu , she is competent. I would like to put both on the same level in terms of captaining their ships.

Armanenets
XM47 "Tristan" dual beam cannon x 2, (Similar to Gottfreid, they're used for anti-ship duels and have better recharge then the Tanhauser)
M10 "Isolde" 42cm triple cannon x 1 (Shell weaponry. this I believe is one of the few advantages Minerva has over AA as AA is still vulnerable to projectile weaponry. AA's only equivalent is the Valiant rail guns, which so far, have not seen deployed effectively against warships)

QZX-1 "Tannhäuser" positron cannon x 1, (Minerva's ultimate weapon. Since it has only one, I think is safe to assume that it has a faster recharge rate the AA's two Lohengrins.

40mm CIWS x 12, missile launchers ("Neidhardt" space missiles,
"Parsifal" ground missiles, "Dispar" interceptor missiles) x many, torpedo launcher ("Wolfram" M25 torpedoes) x 4, anti-beam depth charge launchers x many

Weaknesses: Not much. I would say the lack of submersible capabilities but that is irrelevant to the scenario I've set below. It does seem that AA is slightly more heavily armed than Minerva is you take into account that Minerva has only one Isolde cannon while AA has 2 Valiant rail guns.

First debate

Battleground: Space (assume deep space, no gravity factors like Earth and the Moon to hamp their maneuverabilities.

OK people, throw your opinions. AND MAKE SURE THEY'RE WELL DOCUMENTED AND SUPPORTED WITH ON-SCREEN EVIDENCE. I'm an auditor by profession and i like to see evidence back opinions.