One thing common to all* these sentences is that the inversion of SV to VS allows the sentence to begin with the adverb(ial)—which in all of these examples is temporal: (not) only, rarely, never. Furthermore, the only verbs inverted are auxiliary (do, have, can), not main verbs.

Some sentences can be re-inverted to standard SVO order, but this moves the adverb(ial) after the verb, and makes many of the sentences sound awkward.

Not only does he speak French...
He does not only speak French...

Never have I seen such a...
I have never seen such a...

but

Rarely do we ever get to do...
We do rarely ever get to do...

No sooner had he got a bath than...
He had got a bath no sooner than...

Placing the adverb at the beginning of the sentence greatly changes the emphasis, but require the inversion of subject and auxiliary verb.

Personally, I reserve SV inversion for questions, and avoid constructions like this wherever possible.

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*not all of the sentences in the comments follow this construction:

"Tired I am." and "Such a brilliant pianist was he..." are inversions of subject and complement, not subject and verb. However, "Rarely did he find such a brilliant pianist that..." follows this construction.

"Only if he does this..." begins with a temporal adverb, but does not invert SV.