  We consider the $1s$ Lamb shift in hydrogen and helium ions, a quantity,
required for an accurate determination of the Rydberg constant and the proton
charge radius by means of hydrogen spectroscopy, as well as for precision tests
of the bound-state QED. The dominant QED contribution to the uncertainty
originates from $\alpha^8m$ external-field contributions (i.e., the
contributions at the non-recoil limit). We discuss the two- and three-loop
cases and in particular, we revisit calculations of the coefficients $B_{61},
B_{60}, C_{50}$ in standard notation. We have found a missing logarithmic
contribution of order $\alpha^2(Z\alpha)^6m$. We have also obtained leading
pure self-energy logarithmic contributions of order $\alpha^2(Z\alpha)^8m$ and
$\alpha^2(Z\alpha)^9m$ and estimated the subleading terms of order
$\alpha^2(Z\alpha)^7m$, $\alpha^2(Z\alpha)^8m$, and $\alpha^2(Z\alpha)^9m$. The
determination of those higher-order contributions enabled us to improve the
overall accuracy of the evaluation of the two-loop self-energy of the electron.
We investigated the asymptotic behavior of the integrand related to the
next-to-leading three-loop term (order $\alpha^3(Z\alpha)^5m$, coefficient
$C_{50}$ in standard notation) and applied it to approximate integration over
the loop momentum. Our result for contributions to the $1s$ Lamb shift for the
total three loop next-to-leading term is
$(-3.3\pm10.5)(\alpha^3/\pi^3)(Z\alpha)^5m$. Altogether, we have completed the
evaluation of the logarithmic contributions to the $1s$ Lamb shift of order
$\alpha^8m$ and reduced the overall $\alpha^8m$ uncertainty by approximately a
factor of three for H, D, and He$^+$ as compared with the most recent CODATA
compilation.
