I just played a multiplayer game with a human ally versus AI opponents, and we ran into a number of bugs, some of which are pretty serious.  I will list them here:

1. Discrepancy in Culture / Allegiance for Allies - At one point in the game, I noticed that enemy culture was quickly turning the allegiance of one of my ally's planets.  I let him know, but on his screen, the situation was totally different.  I saw his allegiance dropping fast, and he saw his allegiance stable at 90%.  He saw his culture line going down the phase lane, and all I saw was the enemy's culture line.  Eventually, my game actually told me that my ally LOST THE PLANET, and my game showed the planet with 100% allegiance to the enemy, but for my ally, the planet was still his colony with 90% allegiance!
2. Broken Shared Vision / Ship View - When the game first started, it seemed like shared vision / ship view was working great.  About an hour into the game, I asked my ally when he was going to colonize a planet one jump away from his homeworld.  It turns out he had colonized it 30 minutes prior.  We realized that shared vision / ship view were 100% limited to the first couple of systems that each player colonized!  We could not see each other's colonies or ships as soon as they left the initial planets they colonized!  This makes it very difficult to strategize and work with your ally!
3. Broken Trade Pacts - Once my ally and I had researched some of the Trade Pacts under the Diplomacy research tree, we tried to initiate some trade pacts.  I offered him a pact, and he accepted.  On his screen, it showed the pact as accepted, and showed him getting the bonuses.  On my screen, the pact was permanently waiting for him to respond!
4. False Player Defeats - At various points in the game, both me and my ally were informed that the other person had been defeated.  At one point, my game even showed my ally's homeworld as being owned by an enemy faction, even though he still owned all of his colonies!  This seemed to happen after an enemy bombarded a planet, but failed to win the battle.
5. Crash on Ally Ship Selection - About 10 minutes into the first game we tried, my ally selected one of my ships and zoomed in on it.  His game promptly crashed to the desktop.

1. Research Queue - It would be really great if there was a way to turn on a graphical view of the research queue on the HUD so that you didn't have to hover over the research button to see what was happening.
2. Finished Research - On high-resolution screens, it is very difficult to see the tiny little grey triangle in the bottom left that indicates that something has already been researched.
3. Required Resources Indication - The greying-out of the tech buttons doesn't at all clarify if you can't research something because you lack resources, or if you need more research stations.
4. Toggle Permanent InfoCards - It can be really obnoxious to hover over a ship or the center of the command bar, on large screens, just to see an info card.  There is plenty of screen real-estate to have the info card pop up automatically for whatever the current selection is, and it would make the interface much more useful.  This doesn't need to be something forced on players - the option to enable / disable it seems like a useful feature.
5. Search for Heavy Constructors - In the search menu in the empire bar, it would be really useful if a button was added to find heavy constructors, similar to the button for finding colony ships.
6. Hotkey for Center + Zoom - While the spacebar centers the view on the current ship, it would be nice to have a hotkey that centers the view *AND* zooms in to the current selection.
7. Hotkey to cycle through capital ships - In large fleets, it would be really useful to be able to cycle through capital ships with a simple hotkey to use / manage special abilities.  'Tab' takes you through all of the ships, which can make it very obnoxious to find the capital ships that you might want to manage a little more closely.
8. Battle Updates - With large fleets engaging in multiple battles, it can be difficult to track overall progress across numerous planetary systems.  It would be really useful if there was audio feedback on general fleet statuses - things like enemy reinforcements arriving, the fleet taking heavy losses, planet bombardment started, etc.,

Well, that's it for now!  Any feedback is welcome, especially from Stardock.