  [Abridged] The primordial scalar power spectrum is well constrained on large
scales, primarily by the observations of the anisotropies in the cosmic
microwave background (CMB). Over the last few years, it has been recognized
that a sharp rise in power on small scales will lead to enhanced formation of
primordial black holes (PBHs) and also generate secondary gravitational waves
(GWs) of higher and, possibly, detectable amplitudes. It is well understood
that scalar power spectra with COBE normalized amplitude on the CMB scales and
enhanced amplitudes on smaller scales can be generated due to deviations from
slow roll in single, canonical scalar field models of inflation. In fact, an
epoch of so-called ultra slow roll inflation can lead to the desired
amplification. We find that scenarios that lead to ultra slow roll can be
broadly classified into two types, one wherein there is a brief departure from
inflation (a scenario referred to as punctuated inflation) and another wherein
such a departure does not arise. We consider a set of single field inflationary
models involving the canonical scalar field that lead to ultra slow roll and
punctuated inflation and examine the formation of PBHs as well as the
generation of secondary GWs in these models. Apart from considering specific
models, we reconstruct potentials from certain functional choices of the first
slow roll parameter leading to ultra slow roll and punctuated inflation and
investigate their observational signatures. In addition to the secondary tensor
power spectrum, we calculate the secondary tensor bispectrum in the equilateral
limit in these scenarios. Moreover, we calculate the inflationary scalar
bispectrum that arises in all the cases and discuss the imprints of the scalar
non-Gaussianities on the extent of PBHs formed and the amplitude of the
secondary GWs.
