We consider the four-derivative modification to the Einstein-Hilbert action of general relativity, without a cosmological constant.
Higher derivative terms are interesting because they make the theory renormalisable (but non-unitary) and because they appear generically in quantum gravity theories.

We consider the classical, static, spherically symmetric solutions, and try to enumerate all solution families.
We find three families in expansions around the origin: one corresponding to the vacuum, another which contains the Schwarzschild family, and another which does not appear in generic theories with other number of derivatives but seems to be the correct description of solutions coupled to positive matter in the four-derivative theory.
We find three special families in expansions around a non-zero radius, corresponding to normal horizons, wormholes and exotic horizons.
We study many examples of matter-coupled solutions to the theory linearised around flat space, which corroborate our arguments.

We are assisted by use of a "no-hair" theorem that certain conditions imply that $R=0$, which is applicable in many cases including asymptotically flat space-times with horizons.

The Schwarzschild black hole still exists in the theory, but a second branch of black hole solutions is found that can have both positive and negative mass, and that coincide with the Schwarzschild black holes at a single mass.

The space of asymptotically flat solutions is probed numerically by shooting inwards from a weak-field solution at large radius, and the behaviour at small radius is classified into the families of series solutions (most of which make an appearance).
The results are inconclusive but show several interesting features for further study.