united-states
Suppose that one's spouse has been named in a statement of offence by
a municipal court. If found guilty, the charge would result in a fine
that would ultimately affect both spouses.
The fact that a fine would ultimately affect both spouses is legally irrelevant.
Assume that the spouse that wishes to assist is not a lawyer (but may
have a better grasp of the relevant law than their spouse, be better
at expressing themself in stressful situations, etc.). To what extent
can that spouse aid in the case?
A spouse can provide services similar to a legal secretary or paralegal, such as organizing papers, typing things up from longhand notes or dictation, filling in court forms for final approval by the party spouse, or being present for moral and emotional support. Under Canadian law, what a spouse does is basically limited to what is allowed to be done by a McKenzie Friend.
There is not a consensus legal term for someone doing the same thing in U.S. courts, but all or most of what a McKenzie Friend is allowed to do is also permissible for a non-lawyer to do in a U.S. court case. A McKenzie Friend can assist "by prompting, taking notes, and quietly giving advice." In U.S. Courts, prompting and taking notes for a pro se party are usually O.K., but giving advice to a litigant is usually not O.K.
See also here (discussing whether U.S. law has McKenzie Friends).
Obviously, spouses will discuss things in private; at what point would
such discussions become "legal advice"?
Spousal privilege would prevent a prosecution based upon private communications between spouses, even if it would otherwise be legal advice that is unauthorized practice of law.
If they are talking to certain parties (prosecution, judge, witnesses,
etc.) can the non-party spouse do any of the talking? (assuming the
spouse that is a party to the case would be present and consenting).]
No.
Assume that the costs involved don't warrant hiring a lawyer, and/or
that they couldn't afford one even if they wanted to.
This doesn't impact whether it is unauthorized practice or not.
But, in a case where incarceration is a possibility, there is a right to counsel at state expense if you can't afford one of your own. Scott v. Illinois, 440 U.S. 367, 373–74 (1979). so it shouldn't be as pressing an issue in that situation. If the only potential punishment is a fine, there may not be a right to counsel, unless a state statute or constitution establishes that right. As a practical matter, most municipal ordinance violations carry a possibility of incarceration even if it is rarely, if ever, imposed, but the question assumes that this is not the case in this particular prosecution.