  CONTEXT. The GOSSS spectral classifications and Gaia data have significantly
improved our ability to measure distances and determine memberships of stellar
groups with OB stars. AIMS. We have started a program to identify, measure
distances, and determine the membership of Galactic stellar groups with OB
stars. We start with the identification and distance determinations of groups
with O stars. In this paper we concentrate on groups that contain stars with
the earliest spectral subtypes. METHODS. We use GOSSS to select Galactic
stellar groups with O2-O3.5 stars and a method that combines Gaia DR2
photometry, positions, proper motions, and parallaxes to assign robust
memberships and measure distances. We also include the two clusters in that
paper to generate our first list of 16 O-type Galactic stellar groups. RESULTS.
We derive distances, determine the membership, and analyze the structure of
sixteen Galactic stellar groups with O stars, Villafranca O-001 to O-016,
including the 14 groups with the earliest-O-type optically-accessible stars
known in the Milky Way. We compare our distance with previous results and
establish that the best consistency is with VLBI parallaxes and the worst is
with kinematic distances. Our results indicate that massive stars can form in
relatively low-mass clusters or even in near-isolation, as is the case for the
Bajamar star in the North America nebula. This lends support to the
hierarchical scenario of star formation, where some stars are born in
well-defined bound clusters but others are born in associations that are
unbound from the beginning: groups of newborn stars come in many shapes and
sizes. We propose that HD 64 568 and HD 64 315 AB could have been ejected
simultaneously from Villafranca O-012 S. Our results are consistent with a
difference of approx. 20 microas in the Gaia DR2 parallax zero point between
bright and faint stars. (ABRIDGED)
