We present observations between 14.2 and 17.9GHz of 12 reported supernova remnants (SNRs) made with the Arcminute Microkelvin Imager Small Array (AMI SA). In conjunction with data from the literature at lower radio frequencies, we determine spectra of these objects. For well-studied SNRs (Cas A, Tycho's SNR, 3C58 and the Crab Nebula), the results are in good agreement with spectra based on previous results. For the less well-studied remnants, the AMI SA observations provide higher-frequency radio observations than previously available, and better constrain their radio spectra. The AMI SA results confirm a spectral turnover at ~=11GHz for the filled-centre remnant G74.9+1.2. We also see a possible steepening of the spectrum of the filled-centre remnant G54.1+0.3 within the AMI SA frequency band compared with lower frequencies. We confirm that G84.9+0.5, which had previously been identified as a SNR, is rather an HII region and has a flat radio spectrum. We kindly request that any reference to this paper cites `AMI Consortium: Hurley-Walker et al. 2009'. Issuing author: E-mail: nh313@mrao.cam.ac.uk