Do a Whole Exome Sequencing (WES). It is a cost/efective technique and you can identify mutations of all coding regions of the genome that could affect the phenotype of the cell line.
the best way would be to run WES on affected tissue and compare to health tissue from the same patient. filtering the affected variants by the healthy variants will give you the somatic mutations. an other way if you don't have the healthy counterpart would be to filter with 1000genomes or other database variations.
As mentioned above NGS is a popular approach, however you will need to have some familiarity with bioinformatics and some programming. If the allele fraction is >20% you can opt for sanger.
Alternatively, you can use digital PCR. Article Multiplex Droplet Digital PCR Quantification of Recurrent So...