Magnetic reconnection is a candidate mechanism for particle acceleration in a variety of astrophysical contexts. It is now widely accepted that reconnection plays a key role in solar flares, and reconstructions of coronal magnetic fields indicate that three-dimensional (3D) magnetic null points can be present during flares. We investigate particle acceleration during spine reconnection at a 3D magnetic null point, using a test particle numerical code. We observe efficient particle acceleration and find that two energetic populations are produced: a trapped population of particles that remain in the vicinity of the null, and an escaping population, which leave the configuration in two symmetric jets along field lines near the spine. While the parameters used in our simulation aim to represent solar coronal plasma conditions of relevance for acceleration in flares, the fact that the 3D spine reconnection configuration naturally results in energetic particle jets may be of importance in other astrophysical situations. We also compare the results obtained for the spine reconnection regime with those for the other possible mode of 3D reconnection, fan reconnection. We find that in the latter case energetic particle jets are not produced, though acceleration is observed.