Mnemonics

"hat", "disjunct", "but not", "upto"

-- Used when something "wears many hats"
-- Also used in exclusive or and other disjunctions (you can only wear one hat at a time)

Used alone

^ Before a value, the upto operator (a Bartholomew Cubbins hat)
^ Before a package name is shorthand for introspection (lets you see its hat collection.)
^ In a junction creates a one() junction

Operators containing this character

~^ and +^ and ?^ An exclusive or bitwise operation (or an invert, if only one argument)
^ placed on either or both ends of a range (..) or flip-flop range (ff or fff) to exclude (or "cap") the endpoints
.^ as a method is also for introspection
^^ tight logical exclusive-or

When used inside regexes

^ Matches the start of a string
^^ Matches the start of a line

Special Variables

Variable names starting with hats inside a pointy block or sub are placeholder variables
Method names declared with hats are metaclass methods

Other Uses

Method names starting with hats are metaclass methods

Old, deprecated, or other language uses

^ was bitwise exclusive or in Perl 5. In Perl 6 that's ~^, +^, or ?^ for bitwise, and ^^ in logic.
Lots of special variables in Perl 5 started with ^. Not only are those gone, but placeholders cannot have single uppercase names