  Optical singularities, which are positions within an electromagnetic field
where certain field parameters become undefined, hold significant potential for
applications in areas such as super-resolution microscopy, sensing, and
communication. This potential stems from their high field confinement and
characteristic rapidly-changing field distributions. Although the systematic
characterization of the first singularities dates back many decades, recent
advancements in sub-wavelength wavefront control at optical frequencies have
led to a renewed interest in the field, and have substantially expanded the
range of known optical singularities and singular structures. However, the
diversity in descriptions, mathematical formulations, and naming conventions
can create confusion and impede accessibility to the field. This review aims to
clarify the nomenclature by demonstrating that any singular field can be
conceptualized as a collection of a finite set of principal, 'generic'
singularities. These singularities are robust against small perturbations due
to their topological nature. We underscore that the control over the principal
properties of those singularities, namely, their protection against
perturbations and their dimension, utilizes a consistent mathematical
framework. Additionally, we provide an overview of current design techniques
for both stable and approximate singularities and discuss their applications
across various disciplines.
