4

Dictionaries say that they are synonyms.

Thus my question is what shades of meaning, and in what cases, would you use one over the other? Or, perhaps, are they completely identical in meaning?

5

По счастью tends to mean по счастливой случайности (thanks to unexpectedly good luck) while к счастью is universally used.

Дуэль, по счастью, окончилась благополучно. (А. Куприн, Поединок.)

An even clearer illustration:

По счастью, близко тут журавль случился. (И. Крылов, Волк и журавль.)

3

Stylistic shades are very light.

По счастью is more colloquial and as per my feeling has the undertone of appeal to life circumstances, which aren't purely accidental or miraculous , to greater predictability and expectancy, as it were, of the 'luck' (although Alex_ander here thinks otherwise, so the connotations aren't all that palpable).

Also it seems more often than not to be used with verbs in the Present tense, whereas к счастью tends to accompany verbs in the Past tense.

In the Corpus of Russian language по счастью only gives 965 entries versus 6 907 for к счастью. You're welcome to follow the links to review the actual texts they're used in.

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