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Quantum coherence is an essential feature of quantum mechanics which is responsible for the departure
between the classical and quantum world. The recently established resource theory of quantum coherence
studies possible quantum technological applications of quantum coherence, and limitations that arise if one
is lacking the ability to establish superpositions. An important open problem in this context is a simple
characterization for incoherent operations, constituted by all possible transformations allowed within the
resource theory of coherence. In this Letter, we contribute to such a characterization by proving several upper bounds on the maximum number of incoherent Kraus operators in a general incoherent operation. For a single qubit, we show that the number of incoherent Kraus operators is not more than 5, and it remains an open question if this number can be reduced to 4. The presented results are also relevant for quantum thermodynamics, as we demonstrate by introducing the class of Gibbs-preserving strictly incoherent operations, and solving the corresponding mixed-state conversion problem for a single qubit.