Under what conditions is a monopoly undesirable? First of all, I realize that "undesirable" is an ambiguous term. So, to clarify, when is a monopoly undesirable under the following metrics?


Pareto efficiency
Reduces consumer surplus
Social Welfare (could this possibly be different from the pareto efficiency criterion?)


Are there any criteria that I'm missing? Also, if we take into account potential general equilibrium effects (like the effect on wages), does the analysis change?
Firstly, suppose we take a utilitarian welfare standard that is linear in money. That is to say, suppose that both utility and profits are linear in the amount of money that consumers and firms have (but not necessarily linear in anything else). In that case, the Pareto standard and the utilitarian social welfare criterion coincide exactly! You can see a nice video of Jeff Ely talking about the intuition for this result here (look at the video titled "efficiency"). Intuitively, if both the utility and profits are linear in money then we can always maximise utilitarian welfare by implementing the Pareto optimum and then constructing side-payments to support it.



Now, the answer to the question of when a monopolist is undesirable depends on the richness of the model that one has in mind. In a very basic textbook model of a monopolist, FooBar's criterion is a good one. We know that an ordinary competitive market equilibrium maximises total welfare (the shaded area in the below figure) with price equal to marginal cost (n.b. the supply curve and the marginal cost curve are essentially the same thing):



Since the profit that a monopolist makes from a unit is equal to difference between its price and its marginal cost, the monopolist will tend to set price above marginal cost. This results in a higher price (the green $p$) and a reduction in total welfare (given by the shaded area in this figure) and consumer surplus (the triangle between the price line and the demand curve):





Since this kind of analysis is standard, it makes sense to think of the case in which a monopolist reduces consumer surplus and welfare as being a kind of 'default' and instead ask "when might a monopolist be desirable"? Here are some situations in which having a monopolist might be better than having very intense competition:


As Jyotirmoy Bhattacharya noted in a comment, First Degree Price Discrimination. This describes the situation in which the monopolist can charge each consumer an individualised price that is exactly equal to that consumer's willingness to pay. Thus, the monopolist captures all consumer surplus so that
$$\text{Social welfare}=\underbrace{\text{Consumer surplus}}_{=0}+\text{Producer surplus}=\text{Producer surplus}.$$
Since producer surplus and social welfare coincide, when the monopolist acts to maximise its own surplus it is also maximising social welfare! However, whilst this is good for welfare, it is very bad for consumers. It also requires that the monopolist has enough information about consumers to accurately estimate their willingness to pay, that such discrimination is not illegal, and that consumers are willing to tolerate it (i.e. there is no PR backlash--see, e.g., here).
Large economies of scale. Sometimes the fixed costs of establishing a business are massive. For example, to set up a rail/telephone network requires that you lay train tacks/phone lines across the entire country. Once this cost is incurred, the infrastructure can be used at close-to-zero marginal cost. In this kind of environment it doesn't make sense for society to incur the fixed cost more than once. The infrastructure company is then called a natural monopolist. In these kinds of industries a monopolist is usually allowed to operate and is either (i) owned by the government, or (ii) heavily regulated to ensure that price is not set far above marginal cost.
Network externalities. Some products are only valuable to you is other people also use them. For example, Facebook and telephones (and Stack Exchange!) are only useful because they can be used to contact other people. The more people you can contact, the more valuable to the product becomes. If everybody uses a single monopoly social network then that network will be more valuable than if people are divided across a large number of competing social networks. Network effects are sometimes call demand-side economies of scale.
Innovation and investment. Firms invest in R&amp;D to develop new products (or new production technologies). Why bother? Presumably, firms do this because they anticipate earning a profit selling those new products (or from the reduced cost of the new technology). But their ability to earn this profit will depend on how much competition the firm faces. Thus, one might think that competition should reduce the incentive to innovate by eroding the associated profits. In fact, a famous paper by Aghion and coauthors shows that adding competition first reduces innovation and then, once competition gets fierce enough, increases it again. Thus, if the choice is between a monopolist and a few firms then a monopolist might result in more innovation.
It's perhaps worth mentioning that this last point is closely related to the idea of intellectual property. Since most intellectual property (i.e. creative works like books, songs, or movies; and inventions) are information goods, they can be duplicated or imitated at zero marginal cost. That means that if we allowed competition we should expect there to be many competitors and it will be very hard for anyone (including the original creator) to make any profit whatsoever. In order to ensure that creators and inventors have an incentive to create new works or new inventions, the state therefore grants them a temporary monopoly in the form of either copyright protection or patent protection.




Those are a few of the most common ways in which a monopolist might be preferable to competition. I am sure there are others that don't come to mind right now. The main message is that you have to look at the specific market context to evaluate whether there might be any problems associated with intense competition. Most often the answer will be that there are not, and the usual assumption is therefore that monopolists are undesirable unless we have good reason to think otherwise.