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We compute the (unbounded) Newton-Okounkov body of the Hilbert scheme ofpoints on $\mathbb C^2$. We obtain an upper bound for the Newton-Okounkov bodyof the Hilbert scheme of points on any smooth toric surface. We conjecture thatthis upper bound coincides with the exact Newton-Okounkov body for the Hilbertschemes of points on $\mathbb P^2$, $\mathbb P^1\times\mathbb P^1$, andHirzebruch surfaces. These results imply upper bounds for the effective conesof these Hilbert schemes, which are also conjecturally sharp in the abovecases.

Effective divisors and Newton-Okounkov bodies of Hilbert schemes of points on toric surfaces