  Hierarchical structure formation leads to a clumpy distribution of dark
matter in the Milky Way. These clumps are possible targets to search for dark
matter annihilation with present and future $\gamma$-ray instruments. Many
uncertainties exist on the clump distribution, leading to disputed conclusions
about the expected number of detectable clumps and the ensuing limits that can
be obtained from non-detection. In this paper, we use the CLUMPY code to
simulate thousands of skymaps for several clump distributions. This allows us
to statistically assess the typical properties (mass, distance, angular size,
luminosity) of the detectable clumps. Varying parameters of the clump
distributions allows us to identify the key quantities to which the number of
detectable clumps is the most sensitive. Focusing our analysis on two extreme
clump configurations, yet consistent with results from numerical simulations,
we revisit and compare various calculations made for the Fermi-LAT instrument,
in terms of number of dark clumps expected and the angular power spectrum for
the Galactic signal. We then focus on the prospects of detecting dark clumps
with the future CTA instrument, for which we make a detailed sensitivity
analysis using open-source CTA software. Based on a realistic scenario for the
foreseen CTA extragalactic survey, and accounting for a post-trial sensitivity
in the survey, we show that we obtain competitive and complementary limits to
those based on long observation of a single bright dwarf spheroidal galaxy.
