Approved by Omegadonut
First of all, allow me to thank Destiny_Warrior for starting this thread! I'm going to take it with his agreement to expand the OP and include record streaks.
The Battle Subway, just like the Battle Tower of past games, have you play against trainers in progressively tougher matches during your streak. There are several battle methods (tubelines) you can go through:
Green Line (Single Battle) - Simple 1v1 Battles
Dark Green Line (Super Single Battle) - Difficult 1v1 Battles
Red Line (Double Battle) - Simple 2v2 Battles
Pink Line (Super Double Battle) - Difficult 2v2 Battles
Orange Line (Multi Battle) - Simple double Battle with a friend/NPC against Subway trainers
Yellow Line (Super Multi Battle) - Difficult double Battle with a friend/NPC against Subway trainers
Light Blue Line (WiFi Battle) - Battle against AI controlled WiFi teams (would like confirmation over this)
Since the Green, Red and Orange Lines end once you face the Subway Master(s), I'm going to list the record streaks only for the "difficult" versions. Also, here is some stuff you could find useful:
List of Pokémon used in the Battle Subway (thanks to Team Rocket Elite for gathering the original data, I only changed the 5th gen names for Pokémon/items/moves to their official ones)
Speed tiers in Battle Subway (courtesy of TRE)
BW Battle Subway TOP 10 Single Battle Records:
1. Fluffyflyingpig (105)* -
Whimsicott, Uxie, Drapion
BW Battle Subway TOP 10 Double Battle Records:
1. Fluffyflyingpig (126)* -
Infernape, Latios, Hydreigon, Metagross
BW Battle Subway TOP 10 Multi Battle Records (friend):
BW Battle Subway TOP 10 Multi Battle Records (NPC):
BW Battle Subway TOP 10 Wi-Fi Battle Record:
Notes: * means not finished streak, can continue