In this thread, will discuss about:
1. What the trio of preposition share the similarity?
2. Specific usage of each preposition of time: in, at, and on.

Introduction
In the context of time, the trio have similar function to the prepositions that used for location, except now it is used to indicate time. All of the preposition on this group will be used for two purposes: 1) determining specific time in calendar (the exact location of time) (eg. February, the sixties), it includes more abstract boundaries like season, festival, etc.; 2) telling notion of interval (when a situation happens what interval)(eg. on vacation {which means that he is on the interval of time which he now in that time is doing a vacation}. However, in-depth discussion is needed to tackle on some of the topics that are in-between specific time and period of time (or notion of interval). We will explain them in detail after simpler usages written the uppermost of the section.



a) In
     In is used for explaining larger periods of time (but specific, nevertheless) (eg. century, year, decade, season, month, semester, week, :
          in June, in 2012, in the 1990s, in the 20th century, in the past, in winter
          We will have a vacation in February

b) On
     On is used for explaining dates and days(Friday, pleasant Saturday, my Birthday, New Year's Eve)
          I met Susan at the Vancouver Olympic Games in 2010, on a pleasant Saturday

c) At
     At is used for explaining fraction/subdivision of a day (hours)
          We had lunch together, and at 5:30 sharp she won her competition



b) If you want to tell that something needs more than one day (then creating an interval) to happen or will happen after some period of time (then plunge you into an time interval of waiting), you use IN: I will keep the promise to finish my project in two days (he promises that he will took two days to finish the project); Andy has gone away. He'll be back in a week (= a week from now)



1) Bruckfield (2012) writes "any chosen day should be treated with the preposition ON, as long as it is not added to another day to form a period of time, which must be accompanied with IN.

2) In events in a day (morning, dusk, evening, noon, etc.):
     ON --> interval: Vehicles always honk on the morning of usual working days.
     IN --> period of time: In the morning, we usually eat breakfast made by sausages.
     AT --> specific events in a day: Chicken cockoo can be heard at dawn.
     AT --> specific time in a day: We will have lunch exactly at noon, so be prepared to go downstair and fetch some food down the street.
AT and IN is different by what so called precise time, non-precise time, and presice event of a day (Bruckfield, 2012). In precise time: at is used (for example: noon and midnight); non-precise time: in or (sometimes at  for precise event of a day) is used (night, evening, afternoon, morning); and in precise event of a day: at is used (dawn and dusk)

3) Differences of in and at in an example
      a) See you at night! (means the meetings are planned)
      b) I see you in the night. (means you unexpectedly meet each other during the night).

4) Vacation, Holiday and Weekend
      a) Weekend from restricted definition, means a day when a week ends. When we consider customs from around the world, weekend not always only one day in other countries, usually two days actually. So we think weekend as period of time rather than exact day of time. In British English, at the weekends/at weekend is used, and in American English, on the weekend/or weekend is used. So, it has different rules based on each countries.
      b) Many holidays happen in only one day, but sometimes, in other culture, it happen for more than one day, ranging from two days to eight days. We usually use on to indicate holidays, regardless of how long the holiday actualy takes. But, in British English, at is used especially on one-day holiday like at Christmas, at Easter, etc.
      c) Vacation is similar to holidays in a sense of delimiting time (time that has its own boundaries) that is preprogramed (vacation by ourselves or company and holidays by customs). Vacation is considered a period of time because you are on that time when you're experiencing vacation and not saying that you are needing an interval of time to reach vacation. So you use ON.

5) Do not use at/on/in before last/next/this/every and it is usually safe to leave out on before days.
     a) I'll see you next Friday not *I'll see you on next Friday
     b) I don't work Monday mornings is an alternative of I don't work on Monday Mornings