Correct usage of ‘on’, ‘at’ and ‘in’

Solution 1:

Sorry but your question is really too general to answer.

I'm sure some brave person will rush in where I fear to tread, but this is a complex issue: you need to go and look at an English grammar book to find all the cases. There are lots of exceptions and funny rules and overlapping cases and context dependent cases and so on and so on. You could quite easily use all three:

My mother is on the shopping centre

= e.g. My mother is standing on top of the roof of the shopping centre

My mother is at the shopping centre

= e.g. My mother has gone shopping and I am at home

My mother is in the shopping centre

= e.g. I am standing outside the shopping centre waiting for my mother.

Solution 2:

There aren’t any simple rules per se, and most people, when asked why you use one preposition over another in a particular case, will usually give an explanation by analogy with a more simple example. But it can be hard to invent these analogies if you don’t already know which word to use.

Fortunately there is a good tool you can use when you are wondering which form to use: check for collocates using a corpus, such as the freely-available Corpus of Contemporary American English.

Using COCA, you can search for frequent prepositional collocates before a questionable phrase using [i*], the code for prepositions. For example, if you are unsure whether you should say “in a bus” or “on a bus” you could search for [i*] the bus, and you would get these incidence counts:

1   ON THE BUS     1444
2   OF THE BUS     792
3   TO THE BUS     414
4   OFF THE BUS    340
5   AT THE BUS     283
6   FOR THE BUS    249
7   FROM THE BUS   179
8   IN THE BUS     170

Clearly the most common prepositional collocate for “the bus” is on, although there are curiously some 170 examples of “in the bus”. So you click on them and you see:

• They were huddled in the bus shelter, out of the wind
• Flea Market 8 a.m. -3 p.m. Saturday in the bus parking lot.
• Optimism is a rare quality at 4 a.m. in the bus station in Cleveland.

So far, so good. These are examples where bus is just the beginning of a longer noun phrase to which in can be perfectly reasonably applied. But then:

• they were in the bus trying to help
• Some team members were still in the bus and we were able to get them out.
• The eight men were separated from the women in the bus, and driven away, along with their father's coffin.
• The tourists in the bus didn't seem to give a damn.

What the heck is going on here? I thought that you had to say “on the bus”, and that “in the bus” is wrong. Well, I guess sometimes you can say “in the bus”, although it seems from the contexts given that you would only use “in the bus” when the bus is stationary and can be thought of as a room or building or sorts.

Anyway, you can learn a lot about how to use in, on, and at by searching through a corpus and seeing which preposition native speakers use most frequently with the noun you are wondering about.

Solution 3:

Prepositions like on, at, and in may be governed by the words they modify; that's a common case.
And in that case there is no option except to learn the preposition(s) governed by each word.

But a large number of governed cases are coherent with
the prototype locative semantics of at, on, and in.

As Fillmore describes it in his Deixis Lectures:

The preposition at is said to ascribe no particular dimensionality to the referent of its associated noun; the preposition on is said to ascribe to the referent of its head noun the property of being a line or a surface; and the preposition in is said to ascribe to the referent of its head noun the notion of a bounded two-dimensional or three-dimensional space.

Consider phrases like at the intersection, on the line, on the page, on the wall, in the city, in the kitchen.

Contrast

• at the corner, which means near or in contact with the intersection of two straight lines, or streets
• on the corner, which locates something as being in contact with part of the surface of some angular two-dimensional figure or three-dimensional object
• in the corner, in which the noun corner is used to indicate a portion of three-dimensional space -- in particular, a part of the interior of, say, a room.