UPDATE: 28th June - twenty seven mega-ships now discovered

I have now visited twenty of the documented mega-ships. I will continue to visit the remaining ships as game-time allows. If you hear of any I've missed, please post the details in the comments.

Also, CMD Sektion has created a list of transcripts for all the discovered Generation Ships datalink logs, so if you don't want to fly there but still want to hear the generation ship's stories, view the uplink logs here. I'd recommend visiting the ships though, as seeing the huge derelict ships with the voices of long dead ghosts coming from the audio logs are something no transcript or screenshot can convey.

What are Mega Ships?

The mega-ships are huge, station-size ships that float or fly in deep space. While they have existed in Elite lore for some time they were only added into the Elite:Dangerous game at the 2.3 "Guardians" update.
They come in nine types; Tanker, Cargo, Asteroid Miner, Flight Operations, Prison Ship, Science Vessel, Passenger, Dredger and Generation Ship and those that have been found (so far) are all over populated space and out into deep space. So far nobody has seen a Dredger....
The flight operations type are dock-able (you can land a ship on them) and generally have small and medium pads. As my exploration ship is an Anaconda, I haven't landed on any as yet.
The generation ships were colony ships, launched from the centre of the bubble centuries ago before the FrameShift drive was invented, the crew intended for their children or grandchildren to be the colonists on distant worlds. There were thousands launched, but only a handful have so far been discovered. Those found all appear to be the ones that the colonists died and the mission failed. Those that succeeded may well have founded human colonies we have yet to discover.

Generation Ship Atlas

System: Charick Drift
Ship type: Bulk cruiser
Dockable: No
Location: 3.4Ls from main star. There are five ship log uplink nodes, which you can scan with a datalink scanner from 200m away. These reveal audio logs titled "no brakes" which tell of the crew's fate.
Distance from Sol: 28.6 ly

Naphtha Class Tanker

System: Okinura
Ship type: Tanker
Dockable: Yes
Location: 318Ls from main star
Distance from Sol: 32.5 ly

Hercules Class Bulk Cargo

System: Oduduro
Ship type: Cargo
Dockable: Yes
Location: 854Ls from main star
Distance from Sol: 40.11 ly

Alcatraz Class Prison

System: Milscothach
Ship type: Prison
Dockable: No
Location: 1,385Ls from main star in orbit around Milscothach 1A
Distance from Sol: 42.68 ly

Generation Ship Lycaon

System: Alaunus
Ship type: Bulk cruiser
Dockable: No
Location: Fly from main star to Alaunus 10, then set destination to the Horae system, but stay in supercruise in Alaunus and fly a further 22,000Ls away from Alaunus 10 until you see the generation ship POI on your contacts. There are five ship log uplink nodes, which you can scan with a datalink scanner from 200m away. These reveal audio logs titled "contagion" which tell of the crew's fate.
Distance from Sol: 52.62 ly

Generation Ship Thetis

System: Nefertem
Ship type: Bulk cruiser
Dockable: No
Location: fly 1,934Ls from main star to Nefertem 6A. There are five ship log uplink nodes, which you can scan with a datalink scanner from 200m away. These reveal audio logs titled "madness and ghosts" which tell of the crew's fate.
Distance from Sol: 53.46 ly

Lowell Class Science Vessel

System: Wolf 485A
Ship type: Science vessel
Dockable: No
Location: 9.67Ls from main star
Distance from Sol: 58.75 ly

The Midas

System: Zeus
Ship type: Flight operations
Dockable: Yes
Location: 7.89Ls from main star in orbit around Zeus 1.
Distance from Sol: 62.53 ly

Generation Ship Venusian

System: Kitae
Ship type: Bulk cruiser
Dockable: No
Location: Fly from main star with destination to the Zephyrus system, staying in supercruise in the Kitae system and fly a further 2,500Ls until you see the generation ship POI on your contacts. There are four ship log uplink nodes, which you can scan with a datalink scanner from 200m away. These reveal audio logs titled "boarding party" which tell of the crew's fate.
Distance from Sol: 63.12 ly

Fisher's Rest

System: Aldebaran
Ship type: Flight operations 
Dockable: Yes
Location: 2,513Ls from main star
Distance from Sol: 66.65 ly

Aquarius Class Tanker

System: Maya
Ship type: Tanker
Dockable: No
Location: the tanker is in orbit around Maya 5 an earth-like world with an ice ring, 530Ls from the main star.
Distance from Sol: 72.57 ly

Collinson Class Asteroid Ship

System: Gateway
Ship type: Asteroid miner
Dockable: No
Location: 299Ls from main star
Distance from Sol: 78.62 ly

Bowman Class Science Vessel

System: Asgaa
Ship type: Bulk Cruiser
Dockable: No
Location: 11.9Ls from main star
Distance from Sol: 90.05 ly

Bellmarsh Class Prison

System: Pohnpet
Ship type: Prison
Dockable: No
Location: 242Ls from main star orbiting Pohnpet 2A
Distance from Sol: 92.04 ly

Riker Class Prison Ship

System: Imiutli
Ship type: Prison
Dockable: No
Location: 1.72Ls from main star
Distance from Sol: 95.71 ly

The Harmony

System: Yum Kamcabi
Ship type: Flight operations
Dockable: Yes
Location: 1,235Ls from main star
Distance from Sol: 104.99 ly

Generation Ship Hyperion

System: Lalande 2966
Ship type: Bulk cruiser
Dockable: No
Location: fly 2,300Ls from main star to Lalande 2966 4, then set course toward the Yemaki system and fly a further 7,300Ls until you see the generation ship POI on your contacts. There are five ship log uplink nodes, which you can scan with a datalink scanner from 200m away. These reveal audio logs titled "the promised land" which tell of the crew's fate.
Distance from Sol: 121.53 ly

Generation Ship Odysseus

System: Ross 859
Ship type: Bulk cruiser
Dockable: No
Location: Fly from main star to the secondary star which is 52,000Ls away, when you reach the secondary star target Ross 859 B 1, you should then see the generation ship POI on your contacts. There are five ship log uplink nodes, which you can scan with a datalink scanner from 200m away. These reveal audio logs titled "our new eden" which tell of the crew's fate.
Distance from Sol: 165.98 ly

Generation Ship Pleione

System: Hez Ur
Ship type: Bulk cruiser
Dockable: No
Location: Fly from main star to the fifth planet in the system, when you reach Hez Ur 5 target the Teuten star system for a heading and fly a further 8,403Ls in super-cruise and you should then see the generation ship POI on your contacts. There are four ship log uplink nodes, which you can scan with a datalink scanner from 200m away. These reveal audio logs titled "the chromosome incident" which tell of the crew's fate.
Distance from Sol: 174.01 ly

The Gnosis ^{_{(pronounced "no sis")}}

System: Varati
Ship type: Flight operations
Dockable: Yes
Location: 822Ls from main star.
Distance from Sol: 213.21 ly

Survey Vessel Pandora

System: HIP 21478
Ship type: Unknown
Dockable: Unknown
Location: orbiting 1st planet
Distance from Sol: 289.44 ly

..

Survey Vessel Victoria's Song

System: HIP 17125
Ship type: Unknown
Dockable: Unknown
Location: orbiting 5th planet with purple rings
Distance from Sol: 354.94 ly

..

Overlook

System: HIP 22460
Ship type: Unknown
Dockable: Unknown
Location: [Permit locked system]
Distance from Sol: 362.01 ly

..

The Cete

System: Col 173 Sector LJ-F C12-0
Ship type: Unknown - Professor Melville's mega ship
Dockable: No
Location: find out when I get there...
Distance from Sol: 1,232.09 ly

..

Sagan Class Tourist Ship

System: IC 1805 Sector AV-O C6-6
Ship type: Passenger
Dockable: No
Location: find out when I get there...
Distance from Sol: 6,194.64 ly

..

Zurara

System: Syreadiae JX-F C0
Ship type: Cargo
Dockable: No
Location: find out when I get there...
Distance from Sol: 12,082.88 ly

..

Foster Terminal

System: Coeus
Ship type: Flight operations
Dockable: Yes
Location: find out when I get there...
Distance from Sol: 22,027.98 ly

..