  The occurrence and properties of Type Ia supernovae (SNe Ia) in
single-degenerate binary systems (white dwarf [WD] + nondegenerate companion)
is examined for galaxies of different types, and as a function of redshift. The
rates and characteristics (peak luminosities, expansion velocities of the
ejecta) expected from the explosion of mass-accreting WDs in symbiotic systems
and "helium star cataclysmics" are found to be different from those arising in
another class of candidate systems: cataclysmic-like (contact) systems (CLS),
where a CO WD accretes hydrogen on a thermal time scale from a Roche-lobe
filling main-sequence or subgiant companion. We derive the evolution of the SNe
Ia rate and properties resulting from the thermonuclear explosion of
sub-Chandrasekhar mass WDs in such systems when they detonate a helium layer
accumulated from steady burning of hydrogen at the surface. A fraction of CLS
are believed to form a subset of the observed luminous supersoft X-ray sources
(SSS). Sub-Chandrasekhar explosions from CLS are disfavored in all types of
galaxies at redshifts z > 1. On the other hand, CLS where the WD succeeds to
grow to the Chandrasekhar mass are more likely found in spiral galaxies, and
absent from early-type galaxies. SNe Ia statistics could (if the uncertainties
still involved are reduced) help to discriminate among proposed SNe Ia
scenarios. The range of variation of the characteristics of SNe Ia in the CLS
scenario should be narrower than in symbiotics. The predicted correlation
between peak luminosity and expansion velociy of the ejecta in SNe Ia coming
from these systems is weak. For CLS, the distinction between the
characteristics of SNe Ia respectively arising from sub-Chandrasekhar and from
Chandrasekhar-mass explosions should be sharp, since all sub-Chandrasekhar
