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Energy-momentum localization for the four-dimensional static and spherically symmetric, regular Simpson–Visser black hole solution is studied by use of the Einstein and Møller energy-momentum complexes. According to the particular values of the parameter of the metric, the static Simpson–Visser solution can possibly describe the Schwarzschild black hole solution, a regular black hole solution with a one-way spacelike throat, a one-way wormhole solution with an extremal null throat, or a traversable wormhole solution of the Morris–Thorne type. In both prescriptions it is found that all the momenta vanish, and the energy distribution depends on the mass <i>m</i>, the radial coordinate <i>r</i>, and the parameter <i>a</i> of the Simpson–Visser metric. Several limiting cases of the results obtained are discussed, while the possibility of astrophysically relevant applications to gravitational lensing issues is pointed out.