I am interested in existing algorithms to compute whether a given noncyclic, nonpure cubic extension $K/\mathbb{Q}$ is monogenic or not, and if so, to give me a defining polynomial for the integral power basis $\mathbb{Z}[\alpha]$. I am also interested in general about the following statement: If we are to fix an integer $\Delta$ and count the number of cubic fields having this discriminant, what proportion of these fields are monogenic/nonmonogenic as either $\Delta$ increases in absolute value or the number of number fields having $\Delta$ as a discriminant increases?
1 Answer
In the paper "Computing all power integral bases of cubic fields" (by Gaal and Schulte, published in Mathematics of Computation in 1989) the authors give an algorithm to determine if a cubic field $K/\mathbb{Q}$ is monogenic or not. It boils down to solving a cubic Thue equation, which the authors effectively solve using Baker's linear forms in logarithms.
Apparently, it is expected that 0% of cubic fields are monogenic, although this has not been proven. In the paper "A positive proportion of cubic fields are not monogenic yet have no local obstruction to being so", Alpoge, Bhargava, and Shnidman prove the statement that is the title of the paper. They also show that a positive proportion of cubic fields are locally monogenic (and respectively have no local obstruction to being monogenic, which is a stronger condition). While I don't think this completely answers your second question, I think it's the best that's presently known.

2$\begingroup$ it is mildly surprising to me that locally monogenic does not imply globally! Dunno why... :) Thanks for your scholarship! :) $\endgroup$ Jul 19, 2022 at 0:19

1$\begingroup$ @paulgarrett a Dedekind domain is locally (at all nonzero prime ideals) a UFD but need not be one globally. $\endgroup$– KConradJul 19, 2022 at 4:41

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