This is ... interesting, as it's another one of those examples of how people tended to just "wing it."
The rules are clear in 1e.
A character can only use the armors and weapons permitted by their class. PERIOD.
Among those weapons, they can wield weapons in which they are proficient with no penalty, and weapons that they are allowed (but not proficient) with a penalty.
Character Classes Table II provides the quick list of those weapons that you can use per class.
The Weapon Proficiency Table tells you the initial number of allowable weapons you are proficient in, and the rate that you gain proficiencies.
So, for example, the Magic User-
Allowable Weapons: Dagger, Dart, Staff
At level 1, you get to choose one (1) weapon to be proficient at.
You gain a proficiency every six (6) levels.
So Futzwizz, the Ungainly, starts at level 1 and chooses the staff.
At level 7, he adds a proficiency in darts.
At level 13, he adds a proficiency in his final weapon, the dagger.
And that's it.
This was always well-known, but did cause some ... edge issues. See, e.g., Dragon #56 Sage Advice:
Q. A bard is limited to the use of certain weapons. However, is it possible for a bard to use a weapon he was previously trained in (for instance, a bow), perhaps with a penalty involved?
A. Again, this is a matter simply resolved by realizing the Players Handbook means what it says. No, bards cannot use bows, because that weapon does not appear in the list of weapons permitted to the class. A character who intends to become a bard should make a point of gaining proficiency with at least some of the weapons usable by a bard, in addition to skills with weapons (such as the bow) which the character might prefer to employ during his tenure as a fighter. A bard-to-be might wisely decide to become proficient with bow and arrow, to improve his chances of surviving during his fighter phase. But the use of that weapon is prohibited when the character switches to the thief class, and it can never again be legally employed before or after the character actually becomes a bard.