  We study phase transitions and the nature of order in a class of classical
generalized $O(N)$ nonlinear $\sigma$-models (NLS) constructed by minimally
coupling pure NLS with additional degrees of freedom in the form of (i) Ising
ferromagnetic spins, (ii) an advective Stokesian velocity and (iii)
multiplicative noises. In examples (i) and (ii), and also (iii) with the
associated multiplicative noise being not sufficiently long-ranged, we show
that the models may display a class of unusual phase transitions between {\em
stiff} and {\em soft phases}, where the effective spin stiffness, respectively,
diverges and vanishes in the long wavelength limit at two dimensions ($2d$),
unlike in pure NLS. In the stiff phase, in the thermodynamic limit the variance
of the transverse spin (or, the Goldstone mode) fluctuations are found to scale
with the system size $L$ in $2d$ as $\ln\ln L$ with a model-dependent
amplitude, that is markedly weaker than the well-known $\ln L$-dependence of
the variance of the broken symmetry modes in models that display quasi-long
range order in $2d$. Equivalently, for $N=2$ at $2d$ the equal-time spin-spin
correlations decay in powers of inverse logarithm of the spatial separation
with model-dependent exponents. These transitions are controlled by the model
parameters those couple the $O(N)$ spins with the additional variables. In the
presence of long-range noises in example (iii), true long-range order may set
in $2d$, depending upon the specific details of the underlying dynamics. Our
results should be useful in understanding phase transitions in equilibrium and
nonequilibrium low-dimensional systems with continuous symmetries in general.
