Context. Small linear carbon chain radicals such as C₂ and C₃ act as both the building blocks and dissociation fragments of larger carbonaceous species. Their rotational excitation traces the temperature and density of local environments. However, these homo-nuclear di- and triatomic species are only accessible through their electronic and vibrational features because they lack a permanent dipole moment, and high signal-to-noise ratio data are necessary as the result of their generally low abundances in the interstellar medium (ISM). Aims. In order to improve our understanding of small carbonaceous species in the ISM, we carried out a sensitive survey of C₂ and C₃ using the ESO Diffuse Interstellar Bands Large Exploration Survey (EDIBLES) dataset. We also expanded our searches to C₄, C₅, and the ¹³C¹²C isotopologue in the most molecule-rich sightlines. Methods. We fitted synthetic spectra generated following a physical excitation model to the C₂ (2-0) Phillips band to obtain the C₂ column density (N) as well as the kinetic temperature (T_kin) and number density (n) of the host cloud. The C₃ molecule was measured through its à − $ \tilde X$ (000-000) electronic origin band system. We simulated the excitation of this band with a double-temperature Boltzmann distribution. Results. We present the largest combined survey of C₂ and C₃ to date in which the individual transitions can be resolved. In total, we detected C₂ in 51 velocity components along 40 sightlines, and C₃ in 31 velocity components along 27 sightlines. Further analysis confirms the two molecules are detected in the same velocity components. We find a very good correlation between N(C₂) and N(C₃) with a Pearson correlation coefficient r = 0.93 and an average N(C₂)/N(C₃) ratio of 15.5± 1.4. A comparison with the behaviour of the C₂ diffuse interstellar bands (DIBs) shows that there are no clear differences among sightlines with and without detections of C₂ and C₃. This is in direct contrast to the better-studied non-C₂ DIBs, which have reduced strengths in molecule-rich environments, consistent with the idea that the C₂ DIBs are indeed a distinguishable DIB family. We also identify, for the first time, the Q(2), Q(3), and Q(4) transitions of the ¹³C¹²C (2-0) Phillips band in the stacked average spectrum of molecule-rich sightlines, and estimate the isotopic ratio of carbon ¹²C/¹³C to be 79±8, consistent with literature results. At this stage it is not yet possible to identify these transitions in individual sightlines. Our search for the C₄ and C₅ optical bands was unsuccessful; even in stacked spectra no unambiguous identification could be made.