  We show that as $T\to \infty$, for all $t\in [T,2T]$ outside of a set of
measure $\mathrm{o}(T)$, $$ \int_{-(\log T)^{\theta}}^{(\log T)^{\theta}}
|\zeta(\tfrac 12 + \mathrm{i} t + \mathrm{i} h)|^{\beta} \mathrm{d} h = (\log
T)^{f_{\theta}(\beta) + \mathrm{o}(1)}, $$ for some explicit exponent
$f_{\theta}(\beta)$, where $\theta > -1$ and $\beta > 0$. This proves an
extended version of a conjecture of Fyodorov and Keating (2014). In particular,
it shows that, for all $\theta > -1$, the moments exhibit a phase transition at
a critical exponent $\beta_c(\theta)$, below which $f_\theta(\beta)$ is
quadratic and above which $f_\theta(\beta)$ is linear. The form of the exponent
$f_\theta$ also differs between mesoscopic intervals ($-1<\theta<0$) and
macroscopic intervals ($\theta>0$), a phenomenon that stems from an approximate
tree structure for the correlations of zeta. We also prove that, for all $t\in
[T,2T]$ outside a set of measure $\mathrm{o}(T)$, $$ \max_{|h| \leq (\log
T)^{\theta}} |\zeta(\tfrac{1}{2} + \mathrm{i} t + \mathrm{i} h)| = (\log
T)^{m(\theta) + \mathrm{o}(1)}, $$ for some explicit $m(\theta)$. This
generalizes earlier results of Najnudel (2018) and Arguin et al. (2019) for
$\theta = 0$. The proofs are unconditional, except for the upper bounds when
$\theta > 3$, where the Riemann hypothesis is assumed.
