  We find optimal (up to constant) bounds for the following measures for the
regularity of the Schramm-Loewner evolution (SLE): variation regularity,
modulus of continuity, and law of the iterated logarithm. For the latter two we
consider the SLE with its natural parametrisation. More precisely, denoting by
$d\in(0,2]$ the dimension of the curve, we show the following.
  1. The optimal $\psi$-variation is $\psi(x)=x^d(\log\log x^{-1})^{-(d-1)}$ in
the sense that $\eta$ is a.s. of finite $\psi$-variation for this $\psi$ and
not for any function decaying more slowly as $x \downarrow 0$.
  2. The optimal modulus of continuity is $\omega(s) = c\,s^{1/d}(\log
s^{-1})^{1-1/d}$, i.e. for some random $c>0$ we have $|\eta(t)-\eta(s)| \le
\omega(t-s)$ a.s., while this does not hold for any function $\omega$ decaying
faster as $s \downarrow 0$.
  3. $\limsup_{t\downarrow 0} |\eta(t)|\,\big(t^{1/d}(\log\log
t^{-1})^{1-1/d}\big)^{-1}$ is a.s. equal to a deterministic constant in
$(0,\infty)$.
  We also show that the natural parametrisation of SLE is given by the fine
mesh limit of the $\psi$-variation. Finally, we prove that a particular moment
condition on the increments of a general stochastic process guarantees that the
process attains a certain variation regularity.
