What is the standard error of the sample standard deviation? I read from there that the standard error of the sample variance is
$$SE_{s^2} = \sqrt{\frac{2 \sigma^4}{N-1}}$$
What is the standard error of the sample standard deviation?
I'd be tempted to guess and say that $SE_{s} = \sqrt{SE_{s^2}}$ but I am not sure.
 A: Let  $\mu_4 = E(X-\mu)^4$. Then, the formula for the SE of $s^2$ is:
$$
se(s^2) = \sqrt{ \frac{1}{n}\left(\mu_4 -\frac{n-3}{n-1} \sigma^4\right)}
$$
This is an exact formula, valid for any sample size and distribution, and is proved on page 438, of Rao, 1973, assuming that the $\mu_4$ is finite. The formula you gave in your question applies only to Normally distributed data.
Let $\hat{\theta} = s^2$. You want to find the SE of  $ g(\hat{\theta})$, where $g(u) = \sqrt{u}$.
There is no general exact formula for this standard error, as @Alecos Papadopoulos pointed out. However, one can drive an approximate (large sample) standard error by means of the delta method. (See Wikipedia entry for "delta method").
Here's how Rao, 1973, 6.a.2.4 put it. I include the absolute value indicators, which he incorrectly omitted.
$$
se(g(\hat{\theta})) \approx |g'(\hat\theta)|\times se(\hat{\theta})
$$
where $g'$ is the first derivative.
Now for the square root function $g$
$$
g'(u) = \frac{1}{2\thinspace u^{1/2}}
$$ 
So:
$$
se(s)\approx  \frac{1}{2 \sigma} se(s^2)
$$
In practice I would estimate the standard error by the bootstrap or jackknife. 
Reference: 
CR Rao (1973) Linear Statistical Inference and its Applications 2nd Ed, John Wiley & Sons, NY
