Plane angle and solid angle have:

SI unit of plane angle is radian & Unit of solid angle is steradian.

For dimension of plane angle:

Arc length, $l = r\theta$

$⇒ \theta = \dfrac{l}{r} = \dfrac{\left[{M}^{0}{L}^{1}{T}^{0}\right]}{\left[{M}^{0}{L}^{1}{T}^{0}\right]} = \left[{M}^{0}{L}^{0}{T}^{0}\right]$

Clearly, it is a ratio of two quantities of the same dimension, hence it is dimensionless.

For dimension of solid angle:

Solid angle can be considered as a 3D analogue of a plane angle. It is given by the equation,

$Ω = \dfrac{A}{{r}^{2}}$, where $A$ is the area and $R$ is the radius.

Therefore, its dimensions will be, $= \dfrac{\left[{M}^{0}{L}^{2}{T}^{0}\right]}{\left[{M}^{0}{L}^{2}{T}^{0}\right]} = \left[{M}^{0}{L}^{0}{T}^{0}\right]$, hence it is also dimensionless.