How to unlock ships in FTL?

Image credit: FTL Wiki

The Kestrel Cruiser is the ship that each player will receive at the beginning. It is a pretty decent ship, but its default attachments are not suitable for high tier encounters.

The United Federation - Have 6 races or aliens as crewmembers at any one time.
Full Arsenal - Install all 8 systems and all 3 subsystems at any one time
Tough Little Ship - Fully repair the ship after reaching 1 HP

Layout A - The Kestrel

- 3 Humans
- 8 Reactors
- 1 Burst Laser II and Artemis Missiles

Layout B - Red-Tail

- 2 Humans, 1 Mantis and 1 Zoltan
- 8 Reactors
- 4 Lasers

Layout C - The Swallow

- 2 Humans, 1 Lanius
- 7 Reactors
- Dual Lasers and Ion Stunner

Image credit: FTL Wiki

The Stealth Cruiser is a ship that has no default shield system, but you can install them by using Scrap. This ship is deadly and (as the name suggests) stealthy, making it easier to avoid damage and pack a stealthy punch.

Achievements:

Bird of Prey - Destroy a ship within a single cloak with the Stealth Cruiser (complete the animation)
Phase Shift - Dodge 9 points of damage with the Stealth Cruiser
Tactical Approach - Reach Sector 8 without going to a beacon with environmental danger

Layout A - The Nesasio

- 3 Humans
- 8 Reactors
- Mini Beam and Dual Lasers

Layout B - DA-SR 12

- 2 Humans and 1 Zoltan
- 7 Reactors
- Glaive Beam

Layout C - Simo-H

- 1 Human, 1 Rockman, 1 Slug
- 7 Reactors
- Laser Charger S and Mini Beam

Image credit: FLT Wiki

The Mantis Cruiser excels at close-range combat, and obtaining it can be quite a chore if you don't know what you're doing. You need to obtain it from a guy named KazaaakplethKilik (good luck reading that) and you must be extremely careful not to destroy it in the process!

Take no prisoners! - Kill 20 ships by Sector 6 while using Mantis Cruiser
Avast, ye scurvy dogs! - Kill 5 enemies without taking any damage or losing Crewmembers while using Mantis
Battle Royale - Kill the last enemy on their ship with your last Crewmember while using Mantis Cruiser

Layout A - The Gila Monster

- 3 Mantis, 1 Engi
- 7 Reactors
- Small Bomb and Basic Laser

Layout B - The Basilisk

- 2 Mantis
- 11 Reactors

Layout C - The Theseus

- 1 Engi, 1 Mantis, 1 Lanius
- 8 Reactors
- Crystal Lockdown Bomb and Stun Bomb

Image credit: FTL Wiki

Although it's an odd-looking ship, the Engi is a unique ship that makes use of drones and ion weapons. Its looks are actually deceiving because it doesn't even look like a ship! Layout A is easy to play and fairly good at doing what it's supposed to do, but that cannot be said for Layout B... still, it's mandatory in order to unlock Layout C.

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Robotic Warfare - Equip 3 drones on the Engi Cruiser
I hardly lifted a finger - Kill an enemy ship only with drones while using Engi Cruiser
The guns... They've stopped - Ion 4 enemy systems or subsystems at the same time while using Engi Cruiser

Layout A - The Torus

- 2 Mantis, 1 Human
- 10 Reactors
- Ion Blast II

Layout B - The Vortex

- 1 Engi
- 9 Reactors
- Heavy Ion, Heavy Laser I

Layout C - Tetragon

- 2 Engi and 1 Lanius
- 9 Reactors
- Dual Lasers

Image credit: FTL Wiki

The Federation Cruiser uses Artillery Beams, making it amazing in terms of damage. They also have unique playstyles, and if you like it at first glance, upgrading them and playing them to learn them further only adds to the fun.

Master of Patience - With the Federation Cruiser, use only the Artillery Beam to destroy an enemy ship while taking no damage.
Diplomatic Immunity - Complete 4 blue events before reaching Sector 5 while using the Federation Cruiser
Artillery Mastery - Reach Sector 5 without upgrading the weapons while using the Federation Cruiser

Layout A - The Osprey

- 1 Mantis, 1 Human, 1 Rockman and 1 Engi
- 8 Reactors
- Burst Laser II

Layout B - Nisos

- 1 Human, 1 Slug and 1 Zoltan
- 9 Reactors
- Dual Lasers and Leto Missiles

Layout C - The Fregatidae

- 2 Zoltan, 1 Human and 1 Mantis
- 7 Reactors

Image credit: FTL Wiki

The Slug Cruiser is a unique ship with some rather interesting loadouts.

We're in position! - With the Slug Cruiser, see into every enemy room without using the sensors.
Home Sweet Home - Jump to 30 locations before reaching Sector 8
Disintegration Ray - Kill 3 enemy crew members with one shot from the Anti-Bio Beam while using the Slug Cruiser

Layout A - Man of War

- 2 Slug
- 8 Reactors
- Anti-Bio Beam, Breach Bomb I and Dual Lasers

Layout B - The Stormwalker

- 3 Slug
- 7 Reactors
- Healing Burst and Artemis Missiles

Layout C - Ariolimax

- 3 Slug
- 9 Reactors
- Chain Burst Laser

Image credit: FTL Wiki

The Rock Cruiser is a ship that, unlike its crewmembers, is prone to fire - and with no airlocks, it's entirely up to the crew to put out the fires (if you've played it, then you know it's going to feel like you've had 10 more in the Rock Cruiser than in any other ship).

Is it warm in here? - Have your crew kill a burning enemy on their ship while using the Rock Cruiser.
Defence Drones Don't Do D'Anything! - Destroy an enemy ship with defence drones deployed while using the Rock Cruiser.
Ancestry - Find the secret sector with the Rock Cruiser

Layout A - Bulwark

- 3 Rockmen
- 8 Reactors
- Hull Missile and Artemis Missiles

Layout B - Shivan

- 4 Rockmen
- 8 Reactors
- Heavy Pierce Laser Mk I and Fire Bomb

Layout C - Tektite

- 2 Rockmen, 1 Crystal
- 8 Reactors
- Swarm Missiles and Heavy Crystal Mark I

Image credit: FTL Wiki

One of the most universally-loved ships, the Zoltan Cruisers offer a number of benefits - shield, Halberd beam and so much more!

Shields Holding - Destroy a ship before it pierces through the Zoltan Shield.
Givin' her all she's got, Captain! - Have 29 power in systems at the same time on the Zoltan Cruiser
Manpower - Reach Sector 5 without upgrading the reactor while using the Zoltan Cruiser

Layout A - The Adjudicator

- 3 Zoltan
- 5 Reactors
- Halberd Beam, Leto Missiles

Layout B - Noether

- 3 Zoltan
- 5 Reactors
- 2 Ion Blasts and Pike Beam

Layout C - Cerenkov

- 4 Zoltan
- 2 Reactors
- Ion Charger

Image credit: FTL Wiki

The Lanius Cruiser is fairly easy to unlock, since all you need to do is basically unlock 5 ships in total (or 4 without counting the Kestrel).

Advanced Mastery - Have Hacking, Mind Control and Battery active at the same time.
Scrap Hoarder - Have 600+ Scrap in your ship storage
Loss of Cabin Pressure - Reach Sector 8 without the ship's oxygen level passing 20% (starts after the first jump)

Layout A - Kruos

- 1 Human, 2 Lanius
- 8 Reactors
- Chain Burst Laser and Ion Stunner

Layout B - The Shrike

- 1 Engi, 2 Lanius
- 8 Reactors
- Advanced Flak

Image credit: FTL Wiki

The Crystal Cruiser is a unique ship that looks absolutely stunning and is equally powerful.

Sweet Revenge - Destroy an enemy ship with a shard from the Crystal Vengeance augment.
No Escape - Use the Lockdown Bomb or the Crystal being power to trap 4 enemies in a room while using Crystal
Clash of the Titans - Kill 10 Rock ships with the Crystal Cruiser

Layout A - Bravais

- 2 Human, 2 Crystal
- 8 Reactors
- Crystal Burst Mark I, Heavy Crystal Mark I

Layout B - Carnelian

- 3 Crystal
- 8 Reactors