Can I play the Supporter "Copycat" when it's the only card in my hand?
Yes, you can. Shuffle your deck, then draw cards equal to the number of cards in your opponent's hand.
Source: HeartGold & SoulSilver FAQ (2010-02-11), PUI Rules Team (2010-02-11)

Relevant cards

Note: All applicable cards may not be listed below.
COL1_77
EX15_73
HGSS1_90
SM7_127
SM7_163
EX7_83
WOTCE1_138