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

For dimension of plane angle:

Arc length, $l=r\theta$

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

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,

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

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