Other than the main Campaign (link) and a variety of different 'generic' missions, X:Rebirth contains a number of unique or rare side quests.

Base game:

...

• Craft a Mk5 Engine from dropped shipwreck loot (repeatable)
• "A Healthy Enthusiasm" (craft a Spacefly Egg Collection and exchange for Rare Weapon or a very large reward)
• "A Piece of History" (craft a unique Ancient Relic Collection and exchange for Mk5 Shields or a large reward)
• "Treasure Hunt" (craft a unique Kha'ak Relic Collection and exchange for Mk5 Shields or a large reward)

...

Introduction

There are many types of missions available to the player. Some are the unique or rare side quests described below.

Base game

• Crafting items into rare equipment

  • • SETA construction
    • The shields, engine & weapon below do not require a mission to /craft/ but they do require one to /fit/. These missions are not found in the BBS 

• Crafting missions  

  • • Rare Engine Parts Required
    • Rare Mk5 Engine construction (or large cash reward)
    • A Healthy Enthusiasm
      • Rare Mk3 Weapon (or very large cash reward)
    • A Piece of History and Treasure Hunt
      
      • Rare Mk5 Shields (or large cash reward)
    • X-perimental
      • Playership Jumpdrive*

• Operation Forge Guard

  • • A chain of generic missions for the Argon

  (OL)

  • • (link)

    • Rewards: unique licence (warning about major bug here)

The Teladi Outpost (DLC)

• The Rising Tide
  • • Faction: Urguas Armaments
    • Reward: 5,000,000 Credits (Normal)
  A chain of generic
  • • Difficulty: Medium
    • Can be found by scanning modules of the Haunt at Hissed Syllable/Serpentine Haze/Fields of Opportunity.
      A chain of generic missions for Urguas
  Armaments 
  • • Armaments
      Corporate espionage: scanning, hacking and assassination

Home of Light (DLC)

• Another Gate Open (triggered after travel into one of the vanilla game's systems, after Campaign is completed if that's your start).* (exploration mission offered at the start of new games or after campaign games complete)
• Xenon Invasion (major mission following on from a Ship in Distress generic mission).

*Jumpdrive & 'Another Gate Open' missions will only become available once you've completed the Campaign if you've selected that start; you'll have to return to OL to see the mission briefing in the BBS