I think this article will help many players.
To enable the Battle Isle in single player mode, you must enter the code :

Battle Isle :
Two player: FIRST, GHOST, GAMMA, MARSS, EAGLE, METAN, FOTON, POLAR, TIGER, SNAKE, ZENIT, DONN, VESTA, OXXID, DEMON, GIANT, EUROP

Single player : CONRA, PHASE, ECOTY MOUNT, FIGHT, RUSTY, FIFTH, VESUV, MAGIC, SPACE, VALEY, TESTY, TERRA, SLAVE, NEVER, RIVER, STORM


Battle Isle The Desert :

Single Player Level Passwords :
1 BLOCK 2 WATCH 3 LAGUN 4 BIRMA 5 SERPT 6 RAMBO 07 YUKON 08 POINT 09 FROGS 10 ITALY 11 LINES 12 VARUS 13 SOUND 14 TWEAK 15 NIPON 16 FLAIR 17 ARROW 18 KORSO 19 NOUTH 20 FJORD 21 DONOR 22 LEYES 23 JUMPY 24 WERFT

Two Player Level Passwords :
1 CLOAK 2 LOSAG 3 BOMBS 4 COMET 5 PEARL 6 MIROR 7 ROMEL 8 MAGMA

Battle Isle The Moon :
One Player Two Player
Level Code Level Code Level Code Level Code
1 LUMIT 9 RAMPE 17 EBTAR 1 LUDOS
2 LUNAR 10 RANGG 18 KARST 2 SONNE
3 LUTOF 11 FILMO 19 KANTO 3 SOTEX
4 SONIX 12 FIEST 20 KAROT 4 RASEN
5 SOWYN 13 FINXT 21 KAISR 5 FISCH
6 SOSOO 14 EBENE 22 SYBIL 6 EBTON
7 SONAF 15 EBSYL 23 SFNIX 7 KABEL
8 RACHE 16 EBONY 24 SYNOM 8 SYTAX

Thanks for the nice overview! Two additions:

The Desert has another single-player level:
25 WINIT

The Moon has two secret levels:
25 DIONE (single player)
9 NAIAD (two players)