The sample state for Shari'ah law is the final state where the Prophet Muhammad ﷺ was the ruler. This state was a Theocracy. As he was the only ruler, it fits your definition of Monarchy: undivided rule or absolute sovereignty by a single person (although he never had the title of a King). Formally, it fits to Dictatorship: a form of government in which a ruler wield absolute power; however, I would avoid this expression because it would put the Prophet ﷺ in a row with a lot of bad and rude rulers). Nevertheless, Muhammad ﷺ in his later period as a ruler had also to take decisions as as a statesman. His decisions as a statesman are documented in the Hadith and – together with the Revelation, serve as the source of Shari'ah.
For many centuries, the form of the state was always a monarchy, a Caliphate. Practical jurisprudence was developed under monarchy. The foundation of Shari'ah law has been set in this context, above all in the first centuries H. The religion was bound to the state, so that Islamic rules were applied in the Islamic states.
Other forms of the state appeared only quite recently, when the Shari'ah law system had been established. Traditional Shari'ah law is incompatible with all forms of government other than monarchy, oligarchy, dictatorship and theocracy.
A theocracy like in Iran only fits 12er Shia Islam because Sunnite and Alevite Islam do not have a similar hierarchy in religion. A form of a democracy-inspired theocratic state applying Shari'ah law may be a state where the judges applying Shari'ah are elected by the people and that has no fixed legislation. Government and parliament decide over infrastructure, budget, fees and taxes. A such state never existed and it is not sure that it would work. Shari'ah law would then have to be extended by new juridic traditions.
A multi-cultural and democratic state (with whatever form of democracy, until constitutional monarchy) cannot apply Shari'a law to its full extent because its principle is that laws can be altered by a decision of the parliament or the people. Rules and practice of an Islam-based democracy must have the option to decide which contents of Shari'ah law is applied. Actually, the few democracies as well as most monarchies, dictatorships and oligarchic governments follow this way.
