Idiomatic auxiliary Modifications

Common idiomatic modifiers for auxiliary verbs

Combination	Example	What the 2nd element does	Semantic effect
can easily	This can easily happen.	qualifies the modal assessment	readily possible
can hardly	I can hardly believe it.	restricts the possibility to an extreme degree	near-negation / extreme difficulty
could easily	The plan could easily fail.	qualifies the modal assessment	quite possible
could hardly	That could hardly be called fair.	strongly restricts the possibility	strong negative assessment
could well	That could well happen.	reinforces the modal assessment	plausible possibility
may easily	The plan may easily fail.	qualifies the modal assessment	quite possible
may simply	She may simply be mistaken.	presents one possible explanation as straightforward	straightforward possible explanation
may still	She may still refuse.	indicates that the possibility remains	continuing possibility
may well	She may well be right.	reinforces the modal assessment	strong plausibility
might just	We might just succeed.	limits the possibility to a narrow margin	marginal / uncertain possibility
might simply	He might simply have forgotten.	presents one possible explanation as straightforward	possible explanation
might still	He might still agree.	indicates that the possibility remains	continuing possibility
might well	He might well refuse.	reinforces the modal assessment	plausible possibility
must surely	He must surely know.	reinforces the modal inference	emphatic inference
should probably	They should probably wait.	qualifies the modal assessment	qualified expectation / recommendation
will certainly	He will certainly object.	reinforces the modal prediction	strong prediction
will never	He will never agree.	places an absolute negative restriction on the prediction	emphatic prediction / negation
will probably	She will probably agree.	qualifies the modal prediction	qualified prediction
would certainly	He would certainly object.	reinforces the modal prediction	strong hypothetical prediction
would just as soon	I would just as soon leave.	forms part of a preference construction	preference
would never	She would never do that.	places an absolute negative restriction on the hypothetical prediction	emphatic hypothetical / characteristic negation
would probably	She would probably agree.	qualifies the modal prediction	qualified hypothetical prediction
would rather	I would rather stay.	forms part of a preference construction	preference

Note

“would rather” is the only element that does not modify: “rather” does not modify “would”.