What is a leaky mode of a waveguide? I know that in a waveguide is possible to distinguish between guided modes and radiative modes.
For example, in a planar waveguide, a guided mode is oscillatory inside the core and vanishes exponentially in the cladding while a radiative mode is oscillatory everywhere.
Recently I have read about leaky mode which are defined as

modes having an electric field that decays monotonically for a finite distance in the transverse direction but becomes oscillatory everywhere beyond that finite distance. (Wikipedia)

But I struggle to find an example of these modes in a simple cases as the planar or rectangular ones.
What is an example of a leaky mode?
 A: The easiest way to get modes like these is to have a finite 'barrier' that the light can tunnel through:

Here you have an inner core of glass surrounded by an air gap (which maps, via the quantum mechanical analogy, to a finite potential-energy step) and then surrounded by more glass. Here the inner core is bounded enough to support a guided mode, but that mode is relatively 'big' and it has a substantial evanescent wave in the air surrounding the inner core, which then gets to the outer mass of glass and propagates freely.
In quantum-mechanical language, these are known as 'resonances' of the (unbounded) well.
A: A leaky mode is not a true mode of the waveguide, it is rather a many wavelengths long controlled coupler/radiator along the axis of a waveguide. Let me quote Elliott:"ANTENNA THEORY AND DESIGN", page 453:

"A waveguide mode typically propagates at a phase velocity greater than the speed of
  light. If the waveguiding structure which supports this mode is properly "opened up,'"
  the energy contained in the mode can be leaked to the exterior region, resulting in
  what is called a leaky wave antenna. In practice, one wishes to govern the rate of
  leakage to achieve a desired aperture distribution. With the aperture many wavelengths
  long, the leakage rate is everywhere low and the phase velocity of the leaky
  mode differs but little from the phase velocity of its nonleaky counterpart. As a consequence,
  there is a quasi-uniform progressive phase distribution to the aperture distribution,
  corresponding to the passage of a fast wave over the aperture. Thus such
  structures are also called fast wave antennas."

