Subspace of $\mathcal{L}(V)$ For which vector space the set of non-invertible operators $T\colon V\longrightarrow V$ is a subspace of $\mathcal{L}(V)$?
I know sum of  two  non-invertible matrices is not a non-invertible. That means set of all non-invertible matrices will not form a subspace of any vector space.
Hence, the set of all non-invertible operators is not a subspace of any vector space. But i am confused my assumption is correct or not?

No, your assumption is not correct. Observe that$$\begin{pmatrix}1&0\\0&0\end{pmatrix}+\begin{pmatrix}0&0\\0&1\end{pmatrix}=\begin{pmatrix}1&0\\0&1\end{pmatrix}.$$Therefore, the sum of two singular matrices can be invertible.
Using this example as a model, it is not hard to prove that the answer to your original question is: only when $\dim V=1$.
