Which of the following statements is false about oxygen and sulphur?

To identify the false statement, each option is evaluated against the chemical properties of oxygen and sulphur.

A) Both oxygen ($[\text{He}] 2s^2 2p^4$) and sulphur ($[\text{Ne}] 3s^2 3p^4$) possess two unpaired electrons in their valence p-orbitals. This statement is true.

B) Oxygen rarely exhibits oxidation states beyond $+2$ due to high electronegativity and lack of d-orbitals, whereas sulphur commonly shows $+4$ and $+6$ states. Oxygen does not demonstrate $+4$ or $+6$ oxidation states, making this statement false.

C) At room temperature, oxygen exists as a diatomic gas ($\text{O}_2$), and sulphur is a solid ($\text{S}_8$). This statement is true.

D) The bond dissociation enthalpy of $\text{O–H}$ in $\text{H}_2\text{O}$ is greater than that of $\text{S–H}$ in $\text{H}_2\text{S}$, resulting in higher thermal stability for $\text{H}_2\text{O}$. This statement is true.

The false statement is $\boxed{\text{B}}$.