Molecular gas constitutes the dominant mass component of protoplanetary discs. To date, these sources have not been studied comprehensively at the longest far-infrared and shortest submillimetre wavelengths. This paper presents Herschel SPIRE FTS spectroscopic observations toward 18 protoplanetary discs, covering the entire 450-1540 GHz (666-195 $mu$m) range at R~400-1300. The spectra reveal clear detections of the dust continuum and, in six targets, a significant amount of spectral line emission primarily attributable to $^{12}$CO rotational lines. Other targets exhibit little to no detectable spectral lines. Low signal-to-noise detections also include signatures from $^{13}$CO, [CI] and HCN. For completeness, we present upper limits of non-detected lines in all targets, including low-energy transitions of H2O and CH$^+$ molecules. The ten $^{12}$CO lines that fall within the SPIRE FTS bands trace energy levels of ~50-500 K. Combined with lower and higher energy lines from the literature, we compare the CO rotational line energy distribution with detailed physical-chemical models, for sources where these are available and published. Our 13CO line detections in the disc around Herbig Be star HD 100546 exceed, by factors of ~10-30, the values predicted by a model that matches a wealth of other observational constraints, including the SPIRE $^{12}$CO ladder. To explain the observed $^{12}$CO/$^{13}$CO ratio, it may be necessary to consider the combined effects of optical depth and isotope selective (photo)chemical processes. Considering the full sample of 18 objects, we find that the strongest line emission is observed in discs around Herbig Ae/Be stars, although not all show line emission. In addition, two of the six T Tauri objects exhibit detectable $^{12}$CO lines in the SPIRE range.
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