Why do cell membranes have a lipid bilayer instead of a monolayer?
Why use membranes?
Compartmentalising the cell has lots of advantages and purposes. In Koshland's 2002 essay, compartmentalisation was described as one of the seven fundamental pillars of life. Broadly speaking, membranes create different sets of conditions (chemical and biological) inside the cells. This allows more efficient functioning and advanced catabolic and metabolic processing. More complex and adaptable systems were thusly achieved by the evolutionary process.
Why a bi-layer?
There have been several molecular dynamic simulations such as this one carried out to investigate the biophysicochemistry of spontaneous bilayer assembly. There, lipids start in random orientations. The ordered bilayers we know and love spontaneously assemble in under 100ns. What's actually happening at a molecular dynamics level is the self-association of the hydrophobic lipid tail groups driven entropically by water. In other words the polar (hydrophilic) head-groups &quot;prefer&quot; interacting with the water (called the interfacial region) and the the hydrophobic tail groups &quot;prefer&quot; not interacting with the water. With those two preferences in play, the lipid bilayer forms.

On a smaller scale, micelles also form spontaneously, but this doesn't offer the compartmentalization required for biological functions.

It would appear that the conditions for an ordered bilayer are so favorable that they take precedence over any other orientation or combination. One apparent exception is the archael monolayer. Do see @WYSIWYG's answer that refers to the exceptional extremophiles that use a monolayer. Their comment summarises the monolayer succinctly:

They are technically monolayer but resemble bilayer. Like bilayer connected via a covalent bond. These membrane lipids have two hydrophilic heads and a hydrophobic &quot;torso&quot;.


Image sources

Image 1: http://www.perkepi.com/cell-membrane/
Image 2: https://en.wikipedia.org/wiki/Lipid_bilayer