Many young, massive stars are found in close binaries. Using population synthesis simulations we predict the likelihood of a companion star being present when these massive stars end their lives as core-collapse supernovae (SNe). We focus on stripped-envelope SNe, whose progenitors have lost their outer hydrogen and possibly helium layers before explosion. We use these results to interpret new Hubble Space Telescope observations of the site of the broad-lined Type Ic SN 2002ap, 14 years post-explosion. For a subsolar metallicity consistent with SN 2002ap, we expect a main-sequence (MS) companion present in about two thirds of all stripped-envelope SNe and a compact companion (likely a stripped helium star or a white dwarf/neutron star/black hole) in about 5% of cases. About a quarter of progenitors are single at explosion (originating from initially single stars, mergers, or disrupted systems). All of the latter scenarios require a massive progenitor, inconsistent with earlier studies of SN 2002ap. Our new, deeper upper limits exclude the presence of an MS companion star >8–10 M⊙, ruling out about 40% of all stripped-envelope SN channels. The most likely scenario for SN 2002ap includes nonconservative binary interaction of a primary star initially ≲23M⊙. Although unlikely (<1% of the scenarios), we also discuss the possibility of an exotic reverse merger channel for broad-lined Type Ic events. Finally, we explore how our results depend on the metallicity and the model assumptions and discuss how additional searches for companions can constrain the physics that govern the evolution of SN progenitors.This work is based on observations made with the NASA/ESA Hubble Space Telescope, obtained at the Space Telescope Science Institute (STScI), which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-26555. It is also based in part on observations made with the Spitzer Space Telescope, which is operated by the Jet Propulsion Laboratory (JPL), California Institute of Technology, under a contract with NASA. Support was provided by NASA through grants GO-14075 and AR-14295 from STScI. We thank Andrew Dolphin for his patient advice on how best to implement artificial star tests in Dolphot. E.Z. is supported by a grant of the Netherlands Research School for Astronomy (NOVA). S.d.M. acknowledges support by a Marie Sklodowska-Curie Action (H2020 MSCA-IF-2014, project BinCosmos, ID 661502). A.V.F.'s group is also grateful for generous financial assistance from the Christopher R. Redlich Fund, the TABASGO Foundation, and NSF grant AST-1211916. N.S. is grateful for support from NSF grants AST-1312221 and AST-1515559. R.G.I. thanks the STFC for funding his Rutherford Fellowship under grant ST/L003910/1 and Churchill College, Cambridge, for his fellowship and access to their library