The dimensions of self inductance are:

The self inductance $\mathrm{L}$ of a coil in which the current varies at a rate $\frac{\mathrm{dI}}{\mathrm{dt}}$ is given by:

$\mathrm{e}=-\mathrm{L}\frac{\mathrm{dI}}{\mathrm{dt}}$

where $e$ is the e.m.f. induced in the coil. Now, the dimensions of e.m.f. are the same as those of potential difference, namely, ${\mathrm{ML}}^2{\mathrm{T}}^{-3}{\mathrm{A}}^{-1}$

Now, 

$\mathrm{L}=-\frac{\mathrm{e}}{{\displaystyle \frac{\mathrm{dI}}{\mathrm{dt}}}}$

$\therefore$Dimensions of

 $$\begin{gathered} \left[\mathrm{L}\right]=\frac{\left[\mathrm{e}\right]}{{\displaystyle \frac{\left[\mathrm{I}\right]}{\left[\mathrm{t}\right]}}} \\ =\frac{{\mathrm{ML}}^2{\mathrm{T}}^{-3}{\mathrm{A}}^{-1}}{{\displaystyle \frac{\mathrm{A}}{\mathrm{T}}}} \\ ={\mathrm{ML}}^2{\mathrm{T}}^{-2}{\mathrm{A}}^{-2} \end{gathered}$$