The CPUs of smartphones are becoming multicore with huge RAM and storage to support a variety of multimedia applications in the near future. A MultiStack Parallel (MSP) sorting algorithm is proposed and named MSPSort to support manycore systems. It can be regarded as many threads of single-pivot interleaving block-based Hoare’s algorithm. Each thread performs compare-swap operations between left and right (stacked and interleaved) data blocks. A number of multithreading features of OpenMP and our own optimization strategies have been utilized. To simulate those smartphones, MSPSort is fine tuned and tested on four Linux systems, e.g. Intel i7-2600, Xeon X5670, AMD R7-1700 and R9-2920. Their memory configurations can be classified as either uniform or non-uniform memory access. The statistical results are satisfied compared to parallel-mode sorting algorithms of Standard Template Library, namely Balanced QuickSort and MultiWay MergeSort. Moreover, MSPSort looks promising to be developed further to improve both run time and stability.