Any cabinet that is wider than 36" should not be used for heavy dishes as the shelves may sag over time, this applies to plywood as well as melamine. Plywood is lighter than both MDF and melamine, and the installers like it, however, plywood has more of a tendency to warp. Plywood may last a bit longer than melamine if it gets wet, but not much. It will rot if it is constantly wet. Plywood is an upcharge as well, because good quality cabinet makers don't use Chinese plywood.
Melamine: Melamine is more economical and easier to look after for the kitchen and bathroom cabinet boxes and shelves. Melamine cabinets are made of a good quality fiberboard with a hard melamine surface that is waterproof and therefore will clean up with soap and water, and stains can be cleaned with Clorox wipes. They are the most durable surface and don't require shelf paper. They also do not tend to hold odors from foods or spices, and they are the most cost effective choice.
MDF: (Medium Density Fiberboard). MDF is a dense fiberboard and very heavy, in fact, too heavy for cabinet boxes. MDF doors cost as much as Maple framed doors and they are heavier than Maple doors. Cabinets are not made of MDF because of its heavy weight. Most slab doors are made of MDF and have some veneer or laminate on the surface. Shaker MDF is a choice that allows for the shape of the shaker door without joints which crack when painted on a wood frame door.
The most expensive dining room table from Italy will often be made of MDF and have a beautiful veneer applied to it. MDF is very dense and it is what a slab door is made of with a veneer applied to it. MDF does not warp like solid wood does. This allows the table to open cleanly without sticking.
Many people have been convinced that wood is the only material to build from. This is just marketing hype. Wood has many great qualities but it also has downsides. It is worth learning about materials as they all have good sides for certain applications and they all have downsides too. Think of how old antique cabinets have squeaky doors and drawers that stick. That is because wood warps. :)
Hope this helps you make the decision between Plywood, MDF and Melamine kitchen and bathroom cabinet boxes and doors.