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Heart Gold and Soul Silver: Move Tutors
Move Tutors can teach certain Pokemon select moves. In Heart Gold and Soul Silver the Move Tutors either charge nothing or Battle Points (BP) that are earned by completing challenges in the Battle Frontier.

Ilex Forest: Headbutt
The guy who teaches Headbutt is at the east end of Ilex Forest near a ledge. You can teach your Pokemon Headbutt at any time throughout the game. Headbutt can be used outside of battles to knock Pokemon out of certain trees.
Move Type Style Pwr Acc Cost
Headbutt normal physical 70 100 Free

Blackthorn City: Elemental Beams
This move Tutor is located in the same house as the Move Deleter and Move Relearner in Blackthorn City. These moves can only be tought to starting Pokemon of any region and their evolutions.
Move Type Style Pwr Acc Cost
Frenzy Plant grass special 150 90 Free
Blast Burn fire special 150 90 Free
Hydro Cannon water special 150 90 Free

Frontier Access:
These Move Tutors can be found in the house south of the Pokemon Center in Frontier Access. You need to acquire Battle Points in order to teach your Pokemon these moves.
Move BP
Air Cutter 48 BP
Ancientpower 40 BP
Aqua Tail 40 BP
Block 32 BP
Bounce 32 BP
Bug Bite 32 BP
Dive 40 BP
Earth Power 40 BP
Endeavor 64 BP
Fire Punch 64 BP
Fury Cutter 32 BP
Gastro Acid 32 BP
Gravity 32 BP
Gunk Shot 32 BP
Heal Bell 48 BP
Heat Wave 48 BP
Helping Hand 40 BP
Ice Punch 64 BP
Icy Wind 48 BP
Iron Defense 40 BP
Iron Head 40 BP
Knock Off 40 BP
Last Resort 48 BP
Low Kick 32 BP
Magic Coat 32 BP
Magnet Rise 40 BP
Mud-Slap 32 BP
Ominous Wind 48 BP
Outrage 48 BP
Pain Split 64 BP
Role Play 48 BP
Rollout 32 BP
Seed Bomb 40 BP
Signal Beam 40 BP
Sky Attack 64 BP
Snore 32 BP
Spite 40 BP
String Shot 32 BP
Sucker Punch 40 BP
Super Fang 40 BP
Superpower 48 BP
Swift 40 BP
Synthesis 40 BP
Tailwind 48 BP
Thunderpunch 64 BP
Trick 48 BP
Twister 40 BP
Uproar 48 BP
Vacuum Wave 48 BP
Worry Seed 32 BP
Zen Headbutt 64 BP