The rules implicitly assume that non-two-handed weapons are one-handed
The two-handed property says:
This weapon requires two hands when you attack with it.
For weapons without the two-handed property, I can't find anything in the rules that directly says they are considered one-handed. However, although there is no explicit "one-handed" weapon property, there is strong circumstantial evidence that the designers consider all non-two-handed weapons to be one-handed weapons. First, the Crossbow Expert feat and the ammunition property both which make explicit reference to "one-handed weapons". In addition, the rules for melee attacks say:
A melee attack typically uses a handheld weapon such as a sword, a warhammer, or an axe.
Similarly, the rules for improvised weapons also reference holding the weapon in your hands:
An improvised weapon includes any object you can wield in one or two hands, such as broken glass, a table leg, a frying pan, a wagon wheel, or a dead goblin.
And assuming that all PCs have 2 hands (which is currently the case for all official playable races), the only ways to wield a handheld weapon are with either 1 hand or 2.
Finally, a few official rulings in the Sage Advice Compendium also reference one-handed weapons, such as:
Can my character wield two heavy weapons if she has the Dual Wielder feat? No. The Dual Wielder feat lets you wield two one-handed weapons. Every heavy weapon in the Player’s Handbook also has the two-handed property.
This ruling strongly implies that any non-two-handed weapon is considered a one-handed weapon, or else it wouldn't be eligible for two-weapon fighting. In general, although it's never explicitly stated, the intent seems to be that every weapon either requires 1 or 2 hands to wield, so any handheld weapon that isn't two-handed must be one-handed.
(This answer was adapted from my answer to another question.)