Enzyme kinetics types?
The issue with Michaelis menten curves, is that they presume only one pathway the enzyme can act by. Therefore if an enzyme has multiple pathways the presumptions fail. An example of this is monoamine oxidase, (Ramsay, R. R., Olivieri, A. &amp; Holt, A. An improved approach to steady-state analysis of monoamine oxidases. J. Neural Transm. 118, 1003–1019 (2011).) This paper has the enzyme specific kinetics for monoamine oxidase - the development of the models is incredibly complicated but if you look at the equations you can see how there are multiple pathways. 

They also presume only one substrate, so multi-susbtrate reactions need to have the kinetics modified as such. For that the mechanism are: Ping-pong, ternary- ordered and random, all of which impact on the kinetics.If the concentration of the intermediate complex is presumed not to change with time, then another model is the Briggs-Haldane model, known as quasi-steady state kinetics. 

Then there are the kinetic models for inhibition. Of which the michaelis menten, where the kinetics vary according to how the inhibitors binds. But as is seen with the above MAO paper, can vary depending on the pathways and binding affinities.  If an enzyme is irreversible then the kinetics also differ (Mcdonald, A. G. &amp; Dublin, T. C. Enzymes : Irreversible Inhibition. 1–17 (2012). doi:10.1002/9780470015902.a0000601.pub2). 

It is also worth noting, it is better to use non-linear regression to analyze kinetics as linearizing the data can lead to calculation errors (despite admittedly being the easier and clearer option).