If secularism is defined as a set of political ideas aiming to separate the governance of state from religion (short version), is secularism compatible with Islam? By that, I mean, can Muslims legally (with respect to Islam, not the state) be supportive of the separation of Islam and state and supportive of a state model founded on religiously neutral principles of rational [scientific and materialistic (in a philosophical, not consumerist sense)] social science and morality?
While historically government and courts had been separated in Islam in as much as the ruler made sure that courts were set up and worked but wasn't otherwise involved in jurisdiction, in an Islamic state, the law that the ruler has to uphold is shariah. In particular, not judging and ruling by shariah is considered kufr (see 5:44) - although I think I recall that Ibn Abbas was asked on this and said that it is lesser kufr, i.e. does not make the ruler a non-muslim.
Given that, the idea of secularism is to make sure law and governance is not based on any religion and that religion has no authority in either area. In that far, Islam is not compatible with secularism in the sense that ruling secularly is sinful in Islam, but they are compatible in the sense that the ruler who rules by secular standards does not automatically become a disbeliever. However: I would strongly suspect that a ruler's preferring to rule by secular law rather than by shariah (as a matter of being internally content with ruling by something that goes against shariah) would qualify as a state of the heart that makes him a disbeliever, although I have no reference for this.