Can I use the Supporter "Copycat" when my opponent has no cards in his or her hand?
Yes, you can. Shuffle your hand into your deck, but don't draw any cards.
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