The degree of the differential equation $\frac{\mathrm{d}^2 \mathrm{y}}{\mathrm{d} x^2}+3\left(\frac{\mathrm{~d} y}{\mathrm{~d} x}\right)^2=x^2 \log \left(\frac{\mathrm{~d}^2 \mathrm{y}}{\mathrm{d} x^2}\right)$ is

The differential equation $\frac{d^2y}{dx^2} + 3\left(\frac{dy}{dx}\right)^2 = x^2 \log\left(\frac{d^2y}{dx^2}\right)$ contains a logarithmic function of the second derivative.

Since logarithmic functions cannot be expressed as polynomials in their arguments, this equation is not a polynomial in its derivatives.

Consequently, the degree of this differential equation is undefined.

Final answer: $\boxed{\text{D}}$