Can Alice prove that she *did not* send an unsigned message to Bob? Suppose Alice wants to send messages to Bob. However she doesn't trust Bob. She believes that Bob may publicly disclose a fake message, claiming that it came from Alice. This message will cause the public to have a negative opinion of Alice.
Alice can use signing to prove that a message came from her. However, she cannot prove to the public that the fake message did not come from her. Bob can always claim that the fake message was not signed by Alice because then Bob could publish the signature along with the message and prove to the public that the message came from Alice.
Assuming that the public trusts Bob more than Alice, how can Alice prove that the message is fake? Ideally, she should not be able to prove that the message is fake if she really did send the message to Bob in an unsigned manner.
There is no way to prove that the message is fake without some trusted authority (such as GMail's DKIM) that automatically provides some level of non-repudiation without Alice being able to influence it (Alice can't send a message with DKIM disabled if she's using GMail).
Alternatively, Alice can habitually sign every message and ensure that those who are receiving those messages know that any unsigned message is not to be trusted. This only works if her audience is technically-minded and not easily swayed by emotive language and unfounded claims. Since you say that the community trusts Bob more than Alice, this is really a no-go.
From a cryptographic point of view, there is no way to prove that a message is not authentic as long as the message and signature can be separated in any way whatsoever.