I am interested in existing algorithms to compute whether a given non-cyclic, non-pure 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/non-monogenic as either $\Delta$ increases in absolute value or the number of number fields having $\Delta$ as a discriminant increases?



Algorithm for computing whether a cubic field is monogenic?
In the paper "[Computing all power integral bases of cubic fields](https://doi.org/10.2307/2008731)" (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](https://arxiv.org/abs/2011.01186)", 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.


