Many small-molecule pharmaceutical drugs are covalent enzyme inhibitors. These are often mechanism-based (suicide) inhibitors, and are therefore competitive with the substrate, but don't follow Michaelis-Menten kinetics because they are not at steady-state.
Covalency does not imply irreversibility, however, because some covalent bonds are reversible. There are also non-covalent drugs that have long residence times on the target and are therefore effectively or nearly irreversible.
As far as Michaelis-Menten kinetics of enzyme inhibitors is concerned, in addition to competitive and non-competitive inhibitors, you should also consider uncompetitive and mixed-type inhibitors.
I strongly recommend reading the book "Evaluation of Enzyme Inhibitors in Drug Discovery" by Robert A. Copeland for more details.
Much more than the kinetic mechanism of inhibition goes into determining the drug dosage of an enzyme inhibitor. One important aspect is the potency of inhibition of the target, which may be related to the mechanism. Other extremely important factors are the pharmacokinetics & pharmacodynamics of the drug.
When used for determining the type of enzyme inhibition, the Lineweaver–Burk plot can distinguish competitive, non-competitive and uncompetitive inhibitors. In mixed inhibition, the inhibitor binds to an allosteric site, i.e. a site different from the active site where the substrate binds. However, not all inhibitors that bind at allosteric sites are mixed inhibitors.Mathematically, mixed inhibition occurs when the factors α and α’ (introduced into the Michaelis-Menten equation to account for competitive and uncompetitive inhibition, respectively) are both greater than 1.
The reason why non-competative inhibitors are preferred is that drug binds to allostaric site without dose response compitition with substrate and preferred non reversable one. However, some drugs may be designed for reversible inhibition on enzyme activity.