Context. The Red MSX Source (RMS) survey is an ongoing multi-wavelength observational programme designed to return a large, high-resolution mid-infrared colour-selected sample of massive young stellar objects (MYSOs). We have identified ∼2000 MYSO candidates located within our Galaxy by comparing the colours of MSX and 2MASS point sources to those of known MYSOs. The aim of our follow-up observations is to identify other objects with similar colours such as ultra compact (UC) HII regions, evolved stars and planetary nebulae (PNe) and distinguish between genuine MYSOs and nearby low-mass YSOs. Aims. A critical part of our follow-up programme is to conduct 13CO molecular line observations in order to determine kinematic distances to all of our MYSO candidates. These distances will be used in combination with far-IR and (sub)millimetre fluxes to determine bolometric luminosities which will allow us to identify and remove nearby low-mass YSOs. In addition these molecular line observations will help in identifying evolved stars which are weak CO emitters. Methods. We have used the 15 m James Clerk Maxwell Telescope (JCMT), the 13.7 m telescope of the Purple Mountain Observatory (PMO), the 20 m Onsala telescope and the 22 m Mopra telescope to conduct molecular line observations towards 508 MYSOs candidates located in the 1st and 2nd Quadrants. These observations have been made at the J = 1−0 (Mopra, Onsala and PMO) and J = 2−1 (JCMT) rotational transition frequency of 13CO molecules and have a spatial resolution of ∼20′′−55′′, a sensitivity of T ∗ A 0.1 K and a velocity resolution of ∼0.2 km s−1. We complement these targeted observations with 13CO spectra extracted from the Galactic Ring Survey (GRS), which have a velocity resolution of ∼0.21 km s−1 and sensitivity T ∗ A 0.13−0.2 K, towards a further 403 RMS sources. Results. In this paper we present the results and analysis of the 13CO spectra obtained towards 911 MYSO candidates. We detect 13CO emission towards 780 RMS sources which corresponds to approximately 84% of those observed. A total of 2595 emission components are detected above 3σ level (typically T ∗ A ≥ 0.3 K), with multiple components being observed towards the majority of these sources – 520 sources (∼56%) – with an average of ∼4 molecular clouds detected along each line of sight. These multiple emission features make it difficult to assign a unique kinematic velocity to many of our sample. We have used archival CS (J = 2−1) and maser velocities to resolve the component multiplicity towards 175 sources (∼20%) and have derived a criterion which is used to identify the most likely component for a further 191 multiple component sources. Combined with the single component detections we have obtained unambiguous kinematic velocities for 638 of the 780 MYSOs candidates towards which CO is detected (∼80% of the detections). The 141 sources for which we have not been able to determine the kinematic velocity will require additional line data. Using the rotation curve of Brand and Blitz (1993) and their radial velocities we calculate kinematic distances for all detected components.