In many thermal systems spontaneous mechanical oscillations are generated under the influence of large temperature gradients. Well-known examples are Taconis oscillations in liquid-helium cryostats and oscillations in thermoacoustic systems. In split Stirling refrigerators the compressor and the cold finger are connected by a flexible tube. The displacer in the cold head is suspended by a spring. Its motion is pneumatically driven by the pressure oscillations generated by the compressor. In this paper we give the basic dynamic equations of split Stirling refrigerators and investigate the possibility of spontaneous mechanical oscillations if a large temperature gradient develops in the cold finger, e.g. during or after cool down. These oscillations would be superimposed on the pressure oscillations of the compressor and could ruin the cooler performance.