Clonal hematopoiesis of indeterminate potential (CHIP) refers to the phenomenon where a hematopoietic stem cell acquires fitness-increasing mutation(s), resulting in its clonal expansion. CHIP is frequently observed in multiple myeloma (MM) patients, and it is associated with a worse outcome. High-throughput amplicon-based single-cell DNA sequencing was performed on circulating CD34+ cells collected from twelve MM patients before autologous stem cell transplantation (ASCT). Moreover, in four MM patients, longitudinal samples either before or post-ASCT were collected. Single-cell sequencing and data analysis were assessed using the MissionBio Tapestri® platform, with a targeted panel of 20 leukemia-associated genes. We detected CHIP pathogenic mutations in 6/12 patients (50%) at the time of transplant. The most frequently mutated genes were TET2, EZH2, KIT, DNMT3A, and ASXL1. In two patients, we observed co-occurring mutations involving an epigenetic modifier (i.e., DNMT3A) and/or a gene involved in splicing machinery (i.e., SF3B1) and/or a tyrosine kinase receptor (i.e., KIT) in the same clone. Longitudinal analysis of paired samples revealed a positive selection of mutant high-fitness clones over time, regardless of their affinity with a major or minor sub-clone. Copy number analysis of the panel of all genes did not show any numerical alterations present in stem cell compartment. Moreover, we observed a tendency of CHIP-positive patients to achieve a suboptimal response to therapy compared to those without. A sub-clone dynamic of high-fitness mutations over time was confirmed.