Here is a list of auxiliary verbs (see also modal verbs):

Auxiliary verbAll formsShort affirmative formsShort negative forms
• DO do, does
did

don't, doesn't, didn't
• BEbe
am, is, are
was, were.
am = 'm
is = 's
are = 're
'm not, isn't, aren't
wasn't, weren't.
• HAVE (=haber) have, has
had
have = 've
has = 's
had = 'd
haven't, hasn't
hadn't
• WILL
'llwon't
• WOULD

'd
wouldn't
• CANcan
could

can't, couldn't.
Special written form: cannot (yes, one word only).
• MUSTmust, have to
had to

mustn't
Remember to write the final T in MUST, then add N'T.
• SHALL
'llshan't
• SHOULD

shouldn't
• OUGHT TO

oughtn't
• MAY

mayn't
(It is very rarely used.)
• MIGHT

mightn't

Some verbs are "semiauxiliary" (they do not always follow the rules for auxiliary verbs).
A special case is "HAVE", which has different meanings and can be used in different ways:

Auxiliary verbAll formsShort affirmative formsShort negative forms
• HAVE GOT have, hashave got = 've got
has got = 's got
haven't (got), hasn't (got)
• HAVE (GOT) TO have (got) to, has (got) to
had to
have (got) to = 've got to
has (got) to = 's got to
don't have to, doesn't have to
didn't have to
• NEED need, needs
needn't
• DARE dare, dares
daren't

Notes:

JJCC