To determine the dimensional formula for Planck's constant, we will start by analyzing the given de-Broglie wavelength equation:

$$\lambda = \frac{h}{\sqrt{2 m E}}$$

Here, $$\lambda$$ is the wavelength, $$h$$ is the Planck's constant, $$m$$ is the mass of the particle, and $$E$$ is the energy of the particle.

First, let's derive the dimensional formula for each term involved:

1. Wavelength $$\lambda$$ has the dimensional formula of length $$[L]$$.

2. Mass $$m$$ has the dimensional formula $$[\text{M}]$$.

3. Energy $$E$$ has the dimensional formula of work, which is force times distance:

$$[E] = [F][L] = [\text{MLT}^{-2}][L] = [\text{ML}^{2}\text{T}^{-2}]$$.

Now, let's rewrite the equation in terms of the dimensions:

$$[L] = \frac{[h]}{\sqrt{2 [\text{M}] [\text{ML}^{2}\text{T}^{-2}]}}$$

Simplifying inside the square root:

$$[L] = \frac{[h]}{\sqrt{2 [\text{M}] [\text{M}] [\text{L}^{2}\text{T}^{-2}]}}$$

$$[L] = \frac{[h]}{\sqrt{2 [\text{M}^{2}] [\text{L}^{2}\text{T}^{-2}]}}$$

Since the constants like 2 do not affect the dimensional formula, we can simplify further:

$$[L] = \frac{[h]}{[\text{M L T}^{-1}]}$$

Cross multiplying to solve for the dimensional formula of $$h$$:

$$[h] = [L] [\text{M L T}^{-1}]$$

$$[h] = [\text{M L}^{2} \text{T}^{-1}]$$

Therefore, the dimensional formula for Planck's constant $$h$$ is:

$$\left[\text{ML}^{2}\text{T}^{-1}\right]$$

Hence, the correct option is:

Option C $$\left[\mathrm{ML}^2 \mathrm{~T}^{-1}\right]$$