The determination of isotope ratios of noble-gas atoms has many applications, e.g., in physics, nuclear-arms control, and Earth sciences. For several applications, the concentration of specific noble-gas isotopes (e.g., Kr and Ar) is so low that single-atom detection is highly desirable for a precise determination of the concentration. As an important step in this direction, we demonstrate the operation of a krypton atom trap trace analysis (ATTA) setup based on a magneto-optical trap (MOT) for metastable Kr atoms excited by all-optical means. Compared to other state-of-the-art techniques for preparing metastable noble-gas atoms, all-optical production is capable of overcoming limitations regarding minimal probe volume and avoiding cross-contamination of the samples. In addition, it allows for a compact and reliable setup. We identify optimal parameters of our experimental setup by employing the most abundant isotope 84Kr, and demonstrate single-atom detection within a 3D MOT.