Blowing up a subvariety - what can happen to the singular locus?

I know that blow ups can be used for the resolution of singular points on a variety $X$.

What I need to know is - if I blow-up along some arbitrary subvariety of $X$, what are the possible outcomes for the dimension of the singular locus of the variety? If the subvariety lies outside the singular locus of $X$, then it stays the same, if it is a carefully chosen singular point, it might go down. $\textbf{Can it go up?}$

To be more specific, my variety is a high dimensional hypersurface, and the subvariety I am blowing up is a $\textbf{linear}$ space of much smaller dimension than the singular locus. I don't know if this changes the situation.

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Take $(x^2+y^4+z^6=0)\subset\mathbb{C}^3$. The only singular point is the origin. Blow it up. i.e. put $x=y_1x_1$, $y=y_1$, and $z=y_1z_1$. We get that the strict transform is $(x_1^2+y_1^2+y_1^4z_1^6=0)$. The singular locus contains now the line $x_1=y_1=0$. The dimension went up.