  Single-field slow-roll inflation predicts a nearly scale-free power spectrum
of perturbations, as observed at the scales accessible to current cosmological
experiments. This spectrum is slightly red, showing a tilt $(1-n_s)\sim 0.04$.
A direct consequence of this tilt are nonvanishing runnings $\alpha_s=\mathrm d
n_s/\mathrm d\log k$, and $\beta_s=\mathrm d\alpha_s/\mathrm d\log k$, which in
the minimal inflationary scenario should reach absolute values of $10^{-3}$ and
$10^{-5}$, respectively. In this work we calculate how well future surveys can
measure these two runnings. We consider a Stage-4 (S4) CMB experiment and show
that it will be able to detect significant deviations from the inflationary
prediction for $\alpha_s$, although not for $\beta_s$. Adding to the S4 CMB
experiment the information from a WFIRST-like or a DESI-like survey improves
the sensitivity to the runnings by $\sim$ 20\%, and 30\%, respectively. A
spectroscopic survey with a billion objects, such as the SKA, will add enough
information to the S4 measurements to allow a detection of $\alpha_s=10^{-3}$,
required to probe the single-field slow-roll inflationary paradigm. We show
that only a very-futuristic interferometer targeting the dark ages will be
capable of measuring the minimal inflationary prediction for $\beta_s$. The
results of other probes, such as a stochastic background of gravitational waves
observable by LIGO, the Ly-$\alpha$ forest, and spectral distortions, are shown
for comparison. Finally, we study the claims that large values of $\beta_s$, if
extrapolated to the smallest scales, can produce primordial black holes of tens
of solar masses, which we show to be easily testable by the S4 CMB experiment.
