[mayd]

"I have no spoon." Correct in a situation where exactly one spoon is expected.

"There are no spoons here." Correct in a situation where there could be zero, one or more spoons.

[tsimionescu]

There is no spoon in this room. Equivalent to "There are no spoons in this room".

[yorwba]

The information they communicate about the number of spoons in the room is the same.

The information they communicate about the state of mind of the participants in the conversation is not the same.