Morphological instabilities are common to pattern formation problems such as the
non-equilibrium growth of crystals and directional solidification. Very small perturbations
caused by noise originate convoluted interfacial patterns when surface tension is
small. The generic mechanisms in the formation of these complex patterns are present
in the simpler problem of a Hele-Shaw interface. Amid this extreme noise sensitivity,
what is then the role played by small surface tension in the dynamic formation and
selection of these patterns? What is the asymptotic behaviour of the interface in the
limit as surface tension tends to zero? The ill-posedness of the zero-surface-tension
problem and the singular nature of surface tension pose challenging difficulties in
the investigation of these questions. Here, we design a novel numerical method that
greatly reduces the impact of noise, and allows us to accurately capture and identify
the singular contributions of extremely small surface tensions. The numerical method
combines the use of a compact interface parametrization, a rescaling of the governing
equations, and very high precision. Our numerical results demonstrate clearly that the
zero-surface-tension limit is indeed singular. The impact of a surface-tension-induced
complex singularity is revealed in detail. The singular effects of surface tension are
first felt at the tip of the interface and subsequently spread around it. The numerical
simulations also indicate that surface tension defines a length scale in the fingers
developing in a later stage of the interface evolution.