There are only a few processes in the real-world whose behaviour can be accurately modelled with a continuous uniform distribution (i.e. where all outcomes in a range between a minimum and maximum value are equally likely, and there are no possible outcomes outside of that range). Nevertheless, the distribution is of fundamental importance.
The uniform distribution is key in many applications (especially when applied in the range zero to one), as a sample from it is required (in many algorithms for risk modelling using Monte Carlo simulation) to generate samples from other distributions. When a sample from a uniform distribution is used as an input to a function which inverts any other cumulative distribution, a sample from this distribution can be drawn in a representative way (assuming the sampling from the uniform distribution is representative). The simplest example is actually the use of the If statement in Excel® to create a binomial (yes/no) sampling; this is an intuitive and special case of the inversion of the binomial process. More generally, the combination NOMSINV(RAND()) in Excel will sample a normal distribution and so on (Excel has only a limited set of inverse distributions functions, and this is one reason why add-ins, such as @RISK® or RiskSolver® can be important in practical Excel modelling).
The distribution is sometimes referred to as the “no knowledge” distribution; this term actually shows why is occurrence in real life is so rare – because almost always, one does have some knowledge of a situation. One frequent application is in uncertainty modelling of oil and gas reservoirs, where it is sometimes used as a distribution to represent the position of an oil-water contact point. Of course, further geological tests would yield more information so that the distribution is no longer the most appropriate (and is rather then a Bayesian prior),
Other less practical examples of when he distribution would arise include: the position of a molecule of air in a fixed volume, the point on a car tyre where a puncture would next occur, and the length of time that one may have to wait for a train. For the distribution to apply to each of these situations, implied assumptions need to hold, and it is the validity of these assumptions that can be questioned. In the example concerning the waiting time for a train, one would need to assume that trains arrive in regular intervals but that we have no knowledge of the current time, not of other indicators (sound, wind) that a train is in the process of arriving.