I find $\mathbb R_{\geq 0}$ clumsy (I would never write this on a board when working and I don't often see papers writing functions $f$ defines as $f:\mathbb R_{\geq 0}\rightarrow \mathbb R_{\geq 0}$).
$\mathbb R^+$ seems restrictive, not least if you wish to consider higher dimensions.
I like $[0,\infty)$, but it can be awkward in certain settings such as $f:[0,\infty)\times (0,\infty)\rightarrow \mathbb [0,\infty)$ or
$\left\{E\times[0,\infty)\times (0,\infty)\right\}$
Instead I prefer $\mathbb{\bar R_+}$ for the nonnegative reals and $\mathbb R_+$ for the positive reals. This fits with the notion of closure in $\mathbb R$. (This might not suit those who regularly deal with the extended reals, but given that $\mathbb R$ is so standard, it seems natural to take the closure there.) The function $f: \mathbb{\bar R_+}\rightarrow \mathbb{\bar R_+}$ is then clear and reasonably compact. Moreover, $\left\{E\times\mathbb{\bar R_+}\times \mathbb R_+\right\}$ and $f: \mathbb{\bar R_+}\times \mathbb R_+\rightarrow \mathbb{\bar R_+}$ seem to be substantially easier to read than the interval versions above.
Consistency then dictates that $\mathbb Z_+$ denotes the positive integers and whilst $\mathbb {\bar Z_+}$ is arguably unsatisfactory notation for the nonnegative integers because the closure story no longer applies, I would adopt it in order to be consistent. You could use $\mathbb N=\mathbb Z_+\cup\{0\}$, but that seems worse.
I guess it depends on the problem at hand.
ps. I have also seen $\mathbb R_{++}$ for the positive reals and $\mathbb R_+$ for the nonnegative.