We search for far-infrared (FIR) counterparts of known supernova remnants (SNRs) in the Galactic plane (360° in longitude and b = ± 1°) at 70 - 500 μm with Herschel. We detect dust signatures in 39 SNRs out of 190, made up of 13 core-collapse supernovae (CCSNe), including 4 Pulsar Wind Nebulae (PWNe), and 2 Type Ia SNe. A further 24 FIR detected SNRs have unknown types. We confirm the FIR detection of ejecta dust within G350.1-0.3, adding to the known sample of ̃ 10 SNRs containing ejecta dust. We discover dust features at the location of a radio core at the centre of G351.2+0.1, indicating FIR emission coincident with a possible Crab-like compact object, with dust temperature and mass of Td = 45.8 K and Md = 0.18 M☉, similar to the PWN G54.1+0.3. We show that the detection rate is higher among young SNRs. We produce dust temperature maps of 11 SNRs and mass maps of those with distance estimates, finding dust at temperatures 15 ≲ Td ≲ 40 K. If the dust is heated by shock interactions the shocked gas must be relatively cool and/or have a low density to explain the observed low grain temperatures.