Good ships for flag ship

There's always a tradeoff, so you'll need to tailor your choice to play to your strengths and your usual choice of prey.

Smaller ships have better speed and maneuverability, but they don't have much in the way of cannon space and even with triple hammocks you'll be taking a fairly small boarding crew. You can dance circles around larger ships, but you're going to need to because you'll need to hit them a lot of times to reduce their crew complement before boarding, and they only have to get in one lucky hit to ruin your day.

Larger ships have stronger hulls to take more punishment, bigger crew complements to help you when boarding, and lots of cannon to cripple anything that gets into your range. Downside here is that they tend to be ungainly and anything smaller than you will likely manage to successfully Run Away -- and a ship that escapes means a ship that you're not getting any money from. Likewise, sunk ships can impress the governor, but it doesn't net you anything.

My personal favorite is the Brig of War -- in my opinion, an excellent cross between speed, maneuverability, firepower, and crew complement. Especially when fully upgraded it's fast and quick enough to catch even a sloop unless it's running with the wind from the moment the encounter starts, and enough firepower (and crew complement) to make even a Spanish War Frigate think twice about trying to engage.

Best of all, Captain Kidd sails in one, so it's always possible to encounter at least one of them -- as opposed to the Ship of the Line, which requires seriously annoying non-Spanish factions before it becomes available, and every one of them will be armed to the teeth and gunning for you.

EDIT After giving it some more thought, my main counterargument to the Ship of the Line is that in order to get attacked by one and in order to be able to capture it, you pretty much need to have advanced to the point where nothing else is any kind of threat at all, which limits its utility, whereas the Brig of War can be acquired fairly quickly if you're bold and skilled and will serve you well throughout your entire career of mayhem and plunder...

I find the Brig class a bit too unwieldy, it's not quite up with the real heavy hitters, yet lacks the maneuverability and speed of the Sloop class.

I usually trade the initial ship up to a Sloop of War quickly by taking a pirate raider, escort ship, or pirate hunter, and once they become available, my preference is a Royal Sloop. With 20 cannons and 125 crew (187 with triple hammocks) you can engage any ship. You don't have to worry about accidentally sinking ships with a particular good first salvo, and often can evade almost all enemy fire.

With two Royal Sloops I also have a switch-off flagship when a battle was a closer shave than I wanted, while being fast on the world map but being able to field almost 400 pirates to attack cities.

Sometimes, when I find Henry Morgan with his Large Frigate or an invasion force with one, I get a Large Frigate or Ship of Line additionally, which with Reefed Sails have a very nice turning point as well, and can carry 40 (SOL 48) cannons, and up to 250 (SOL 300) crew by themselves outgunning even war galleons.

Shadur's answer is great: there's a tradeoff between being able to catch fast ships and evade fire, and being able to bring a large crew aboard and deal and endure damage.

There's one more thing to consider, though: higher difficulty levels favour certain ship features in combat. On Rogue and especially Swashbuckler the enemies get a substantial speed boost relative to yours, so it becomes harder to outmanoeuvre enemy ships, particularly Sloops or Pinnaces.

On difficulty levels up to Adventurer the Brig of War offers an awesome balance of features, but on harder settings it's a bit less capable of catching ships or dodging cannonballs (though still very much usable). The Royal Sloop is still fast although you'll need to make sure to dance around broadsides. The Mail Runner even more so, but with the smaller crew size it might require some extra fencing skill. And if you're skilled with reefing sails and pre-battle positioning you can use a Frigate class ship without worrying about dodging each cannonball.

More detailed comparison

In my experience, massive crew complement when boarding isn't as important a factor as it may seem to be. Around 150 men rarely get severely outnumbered and can be fitted even in an upgraded Sloop of War, and one or two Grape Shots from almost any ship can level the playing field if needed. Cannon capacity too is a double-edged sword – unless you want to sink the enemy ship (which is almost never), a massive broadside may become a liability as you'll unnecessarily cripple and reduce the price of the captured vessel, instead of merely slowing it down for the combat.

Also, while square rigging makes Frigates quick running before the wind, it leaves them slower than other ships when sailing upwind, which means that they're not that good leading your fleet around the Caribbean. Often there are non-Frigate/Galleon ships in your fleet too, further reducing the top speed advantage.

While these points speak against the Frigate class ships, their durability, crew size and speed in combat still make them formidable. (Those attributes are actually common to all Frigates, so you could argue that the Ship of the Line doesn't hold much advantage over its siblings after all.) And as Shadur noted, at the other extreme the small ships' low firepower prolongs battles when you need to reduce the enemy crew, increasing the risk of eventually catching a broadside.

Ultimately, I find the Royal Sloop and the Brig of War as flagships generally superior (if slightly) to the others. Fast around the map and balanced in combat, they hold up well in all situations and difficulty levels.

As an aside, you might want to have two similar ships sharing the flagship designation even if you don't need the extra crew or cargo capacity. If you lose one ship in battle (which can happen quite suddenly and unexpectedly on Swashbuckler) and don't like save scumming, you can just carry on pirating without having to compromise while you get a new one.