Prepositions of place and time follow a picture you can learn in an afternoon: at is a point with no size, on is a line or a surface, in is something with an inside. Everything outside place and time is a different kind of object, because those prepositions arrived attached to the word in front of them and have to be learned as part of it.
Knowing which of the two you are looking at saves years. If the preposition is telling you where or when, work it out from the picture. If it is sitting behind a verb or an adjective, stop working and learn the pair, because there is nothing under it to work out.
Eight patterns, and the one that is actually systematic
Most prepositions are learned attached to a word. Place and time are the exception and follow a real pattern worth learning once.
- in a country, on a street, at a number. The container narrows: in a large area, on a line, at a point. In Ireland, on Grafton Street, at number 12.
- in the morning, on Monday, at six. The same narrowing applied to time: a stretch, a day, a point.
- depend on, interested in, good at. Fixed, and learned attached to the verb or adjective rather than derived.
- listen to and look at. Both take a preposition where the equivalent verb in many languages takes none, and dropping it is a persistent error.
- arrive in and arrive at. In for a town or country, at for a building or event, and never arrive to, which is the standard Romance error.
- by car and on foot. Transport takes by with no article, except walking, which is the exception everybody meets first.
- different from is safest, with than common in America and to common in Britain, and all three defensible.
- in time and on time. In time means early enough for something; on time means at the appointed moment.
The picture, and the exact place it cracks
The same three words run place and time on the same logic, which is the part that makes them worth learning as a set rather than as two lists. A point in space is at the door and a point in time is at six. A surface is on the table and a day is treated as one, hence on Monday. An enclosure is in the box and a month is treated as one, hence in June. The scale is the same in both: a point sits on a surface, a surface sits inside an enclosure, and the preposition tells you which of the three you have zoomed to.
In the car and on the bus is the crack every learner finds. The conventional account is that a car encloses you and a bus lets you stand and walk about, so public transport takes the surface preposition. Treat that as a convention rather than a rule, because it is one: the picture is a strong first guess and not a law, and pretending otherwise is what makes a learner distrust the whole of it the first time it fails.
Where it stops: still looking for a rule in the third group
The stall is not any particular preposition. It is a learner going into the fifth year of English still asking why depend takes on, why interested takes in and why good takes at, and treating each answer as a rule that failed to arrive. There is no rule. The preposition is part of the entry for the word before it, in the same way that a gender is part of the entry for a noun in German, and the question that would explain it has no answer.
These survive longer than any other error in this cluster because none of them blocks anything. Discuss about the problem is understood on the first pass by every listener alive, so nobody stops you, and after enough repetitions it stops being an error you make and becomes the way you say it. What that costs, and why nothing corrects it, is the errors that survive years of study, and why they stick.
Seeing the pair before you learn it
Reverso Context is the fastest way to find which preposition a word keeps company with. Type the word and it returns real sentences with the phrase highlighted, so interested comes back sitting next to in a hundred times and the pair is visible before you have read a definition. It is free with advertising, and it replaced the bilingual dictionary entry, which hands you a word on its own and is silent about the phrase the word actually lives in.
Its weakness is worth knowing. The sentences come from subtitles, official documents and web text, so the register swings violently between a film script and a European directive, and a phrase that looks common may be common in only one of those. Use it to find the preposition, not to judge how formal the sentence is.
What it looks like when it goes wrong
The recognisable ones are a short list and they nearly all come from the first language having chosen differently. I arrived to London, where English wants in for a city and at for a station. Discuss about and explain me this, where English wants no preposition in the first and to in the second. Married with, where English says to. Depend of, from a language that uses its own version of of there.
Every one of those is a pair remembered from another language rather than a picture applied wrongly, which is why reading more rules will not touch them and why they belong on a list you rehearse. The unstressed small words that behave the same way, arriving with something rather than being chosen, are a, an, the, or nothing at all, and how to tell which, and both sit under the rules that decide what a sentence is allowed to mean.
What follows from this is a question about what you have been storing all along. If the preposition belongs to the word, then the thing worth learning was never the word on its own but the word with its preposition attached, and that changes what a vocabulary entry should look like: building the word count you need to say what you mean.