be, feel, seem, sound
I feel hopeful that a peaceful solution will be found.

become, get
Don't get too hopeful. We may not be able to go.

remain
The police remained hopeful that she would be found alive.

extremely, fairly, very, etc.
almost

about
She was not very hopeful about her situation.

of (BrE)
The police are hopeful of catching the thieves.

be, look, seem
Things aren't looking very hopeful at the moment.

extremely, fairly, very, etc.
There are some fairly hopeful signs of recovery in the US market.

Hopeful is used with these nouns:
beginning, future, note, sign