Borel summable semiclassical expansions in 1D quantum mechanics are considered. These are the Borel summable expansions of fundamental solutions and of quantities constructed with their help. An expansion, called topological, is constructed for the corresponding Borel functions. This allows us to study the Borel plane singularity structure in a systematic way. Examples of such structures are considered for linear, harmonic and anharmonic potentials. Together with the best approximation provided by the semiclassical series the exponentially small contributions completing the approximation are considered. A natural method of constructing such exponential asymptotics based on the Borel plane singularity structures provided by the topological expansion is developed. The method is used to form the semiclassical series including exponential contributions for the energy levels of the anharmonic oscillator.