  Residual errors in shear measurements, after corrections for instrument
systematics and atmospheric effects, can impact cosmological parameters derived
from weak lensing observations. Here we combine convergence maps from our suite
of ray-tracing simulations with random realizations of spurious shear. This
allows us to quantify the errors and biases of the triplet
$(\Omega_m,w,\sigma_8)$ derived from the power spectrum (PS), as well as from
three different sets of non-Gaussian statistics of the lensing convergence
field: Minkowski functionals (MF), low--order moments (LM), and peak counts
(PK). Our main results are: (i) We find an order of magnitude smaller biases
from the PS than in previous work. (ii) The PS and LM yield biases much smaller
than the morphological statistics (MF, PK). (iii) For strictly Gaussian
spurious shear with integrated amplitude as low as its current estimate of
$\sigma^2_{sys}\approx 10^{-7}$, biases from the PS and LM would be unimportant
even for a survey with the statistical power of LSST. However, we find that for
surveys larger than $\approx 100$ deg$^2$, non-Gaussianity in the noise (not
included in our analysis) will likely be important and must be quantified to
assess the biases. (iv) The morphological statistics (MF,PK) introduce
important biases even for Gaussian noise, which must be corrected in large
surveys. The biases are in different directions in $(\Omega_m,w,\sigma_8)$
parameter space, allowing self-calibration by combining multiple statistics.
Our results warrant follow-up studies with more extensive lensing simulations
and more accurate spurious shear estimates.
