Learn English – Use of “last year” and “last one year”

Yes.

It’s very tempting to leave the answer at that, but… some more notes.

1. The most cursory look at any newspaper, or around the blogosphere, confirms this answer. ‘Last week’ is common as muck, while ‘yester-week’ is virtually non-existent, and when it does appear, it’s for archaic effect.

2. ‘Last week’ isn’t in any way incorrect. It’s had the meaning ‘most recent, latest’ for over 600 years: in 1411, for instance, the rolls of Parliament under Henry IV refer to “The last parlement…” (OED.)

3. As your sources point out, however, ‘last’ can sometimes be ambiguous, and ‘latest’ is often a good alternative in those cases. Talking about ‘the latest week’ sounds odd and affected, just as it did when Fowler wrote; but talking about ‘Madonna’s latest single’ is perfectly fine.

4. ‘Yester-’ isn’t actually as archaic as we tend to think! A few examples — ‘yesterday’, ‘yesternight’, ‘yester-eve’ — are old, but (according to the OED again) the general combining form (in eg ‘yester-week’) only really shows up in the 19th century. Amazingly, even good old ‘yester-year’ was apparently only coined in 1870, by Rosetti translating Villon: “Where are the snows of yester-year?”

5. Finally, none of the sources you give support your implication that ‘yester-week’ is the correct usage that should be preferred to ‘last week’! The Bierce and Fowler quotes point out the illogic and occasional ambiguity of ‘last’ (note that neither of these makes it grammatically wrong); Fowler’s suggests ‘latest’ as a sometimes better alternative, but neither even mentions ‘yester-’! The OED simply points out that ‘yester-’ exists (at least, existed for a few decades) and means roughly the same thing.

In the last year, last year, in the past year

How do such things come into being? We may assume that the beginnings were something like this: When was it? - That was in the last year. Then you may assume that this was shortened to: That was last year. The omission of self-evident "in" does not change the meaning. Then "last year" alone can be used as an adverbial group in a sentence. So "Last year we went to Italy" will be the most frequent formula, simply because it is the shortest indication of "when". Of course you can use the longer formula "In the last year we went to Italy." There is no difference of meaning at all. It is only longer, and an author may use it simply for rhythmic and stylistic reasons. You may also use it in spoken language, for example when you need some time to formulate what you want to say. But we can say when there is a short formula then it is the short formula that is most frequently used. In the past year: The idea is exactly the same. It is simply a different "picture". When you say "the last year" you think of a row of things and you choose the thing at the end. When you say "the past year" you think that an event has gone by or passed. If I would look at Google N-gram Viewer, I think, that "in the past year" is less common than "last year". I would say it is a variant that can be found occasionally.