The political instability that is most usually connected to the Great Depression happened in Germany. There, the political instability was caused largely by Germany's dependence on the United States at a time when the US economy was crashing.
Germany's economy depended on exports and on loans from the United States. Germany had to pay reparations for WWI and needed those loans in order to be able to do so. When the Depression hit, exports and loans both dried up. Because Germany's economy was falling apart, people were very upset with the government. This led to great political upheaval and the eventual rise of the Nazi Party.