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Recent study in K-stability suggests that klt singularities whose localvolumes are bounded away from zero should be bounded up to specialdegeneration. We show that this is true in dimension three, or when the minimallog discrepancies of Koll\'ar components are bounded from above. We conjecturethat the minimal log discrepancies of Koll\'ar components are always boundedfrom above, and verify it in dimension three when the local volumes are boundedaway from zero. We also answer a question of Han, Liu and Qi on the relationbetween log canonical thresholds and local volumes.

On boundedness of singularities and minimal log discrepancies of Koll\'ar components