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As you said, analytic sets (other than the whole space) are pluripolar. By playing with subharmnic functions of the form $\sum u_j$ you can arrange some more complicated sets which are neither open nor closed (but the set where a plurisubharmonic function is $-\infty$ must be $G_\delta$). So you can have a dense pluripolar set, for example. Other non-analytic examples can be made by taking subsets of pluripolar sets. Pluripolar sets in (complex) dimension $1$ are easier to visualize: they are just very small, and sufficient conditions for a set to be pluripolar can be given in terms of Haurdorff-like measure terms. Then you can take products of these one dimensional sets times $C^k$.

But there is the following remarkable theorem:

If a pluripolar set in $C^{n+1}$ is a graph of a continuous function $f:C^n\to C$, then this function is analytic (thus the set is also analytic).

Nikolay Shcherbina, Pluripolar graphs are holomorphic. Acta Math. 194 (2005), no. 2, 203–216.

This answers your question about smooth hypersurfaces.

As you said, analytic sets (other than the whole space) are pluripolar. By playing with subharmnic functions of the form $\sum u_j$ you can arrange some more complicated sets which are neither open nor closed (but the set where a plurisubharmonic function is $-\infty$ must be $G_\delta$). So you can have a dense pluripolar set, for example. Other non-analytic examples can be made by taking subsets of pluripolar sets. Pluripolar sets in (complex) dimension $1$ are easier to visualize: they are just very small, and sufficient conditions for a set to be pluripolar can be given in terms of Haurdorff-like measure terms (but only sufficient or necessary, there is a theorem that says that a complete characterization in metric terms is impossible). Then you can take products of these one dimensional sets times $C^k$.

But there is the following remarkable theorem:

If a pluripolar set in $C^{n+1}$ is a graph of a continuous function $f:C^n\to C$, then this function is analytic (thus the set is also analytic).

Nikolay Shcherbina, Pluripolar graphs are holomorphic. Acta Math. 194 (2005), no. 2, 203–216.

This answers your question about smooth hypersurfaces.

As you said, analytic sets (other than the whole space) are pluripolar. Countable unions of analytic sets are pluripolar too. By playing with subharmnic functions of the form $\sum u_j$ you can arrange some more complicated sets which are neither open nor closed (but they must be $G_\delta$). Other non-analytic examples can be made by taking subsets of pluripolar sets. Pluripolar sets in (complex) dimension $1$ are easier to visualize: they are just very small, and sufficient conditions for a set to be pluripolar can be given in terms of Haurdorff-like measure terms. Then you can take products of these one dimensional sets times $C^k$.

But there is the following remarkable theorem:

If a pluripolar set in $C^{n+1}$ is a graph of a continuous function $f:C^n\to C$, then this function is analytic (thus the set is also analytic).

Nikolay Shcherbina, Pluripolar graphs are holomorphic. Acta Math. 194 (2005), no. 2, 203–216.

This answers your question about smooth hypersurfaces.

As you said, analytic sets (other than the whole space) are pluripolar. By playing with subharmnic functions of the form $\sum u_j$ you can arrange some more complicated sets which are neither open nor closed (but the set where a plurisubharmonic function is $-\infty$ must be $G_\delta$). So you can have a dense pluripolar set, for example. Other non-analytic examples can be made by taking subsets of pluripolar sets. Pluripolar sets in (complex) dimension $1$ are easier to visualize: they are just very small, and sufficient conditions for a set to be pluripolar can be given in terms of Haurdorff-like measure terms. Then you can take products of these one dimensional sets times $C^k$.

But there is the following remarkable theorem:

If a pluripolar set in $C^{n+1}$ is a graph of a continuous function $f:C^n\to C$, then this function is analytic (thus the set is also analytic).

Nikolay Shcherbina, Pluripolar graphs are holomorphic. Acta Math. 194 (2005), no. 2, 203–216.

This answers your question about smooth hypersurfaces.

As you said, analytic sets (other than the whole space) are pluripolar. Countable unions of analytic sets are pluripolar too. By playing with subharmnic functions of the form $\sum u_j$ you can arrange some more complicated sets which are neither open nor closed. But there is the following remarkable theorem:

If a pluripolar set in $C^{n+1}$ is a graph of a continuous function $f:C^n\to C$, then this function is analytic (thus the set is also analytic).

Shcherbina, Nikolay Pluripolar graphs are holomorphic.

So nice pluripolar hypersurfaces are analytic. Acta Math. 194 (2005), no. 2, 203–216.

As you said, analytic sets (other than the whole space) are pluripolar. Countable unions of analytic sets are pluripolar too. By playing with subharmnic functions of the form $\sum u_j$ you can arrange some more complicated sets which are neither open nor closed (but they must be $G_\delta$). Other non-analytic examples can be made by taking subsets of pluripolar sets. Pluripolar sets in (complex) dimension $1$ are easier to visualize: they are just very small, and sufficient conditions for a set to be pluripolar can be given in terms of Haurdorff-like measure terms. Then you can take products of these one dimensional sets times $C^k$. 

But there is the following remarkable theorem:

If a pluripolar set in $C^{n+1}$ is a graph of a continuous function $f:C^n\to C$, then this function is analytic (thus the set is also analytic).

Nikolay Shcherbina, Pluripolar graphs are holomorphic. Acta Math. 194 (2005), no. 2, 203–216.

This answers your question about smooth hypersurfaces.