We carry out the deepest and broadest search for continuous gravitational-wave signals with frequencies between 20-1500 Hz, from three neutron stars at the center of the supernova remnants Cassiopeia A, Vela Jr., and G347.3-0.5. This search was made possible by the computing power shared by thousands of Einstein@Home volunteers. After the initial Einstein@Home search, we perform a multi-stage follow-up of the most promising $\approx$ 45 million signal candidates. In the last stages, we use independent data to further investigate the remaining candidates from the previous stages. We set the most stringent constraints to date on the gravitational-wave amplitude, equatorial ellipticity, r-mode saturation amplitude, and -- for the first time -- the neutron-star crustal anisotropy. For spin periods lower than 2 ms we constrain the ellipticity to be smaller than $4\times 10^{-7}$ for all targets. We exclude the crustal anisotropy to be smaller than $5\times 10^{-3}$ for spin periods between 1.3-100 ms. Only one candidate -- from the low frequency G347.3 search -- survives all follow-ups. We illustrate properties of this candidate. Investigations on new data will aid in clarifying its nature. Such ``new" data exists and would be optimal for this purpose, but they are not publicly accessible at the time of writing.

Targeted searches reveal stringent constraints on properties of neutron stars in supernova remnants through Einstein@Home's volunteer computing power.