The instruction set of the 8051 microcontroller is classified into several categories based on the type of operations performed. These include data transfer instructions, byte-level logical instructions, arithmetic instructions, bit-level instructions, rotate and swap instructions, and jump and call instructions. Data transfer instructions move data between registers, memory, and I/O ports using various addressing modes. Byte-level logical instructions perform AND, OR, and XOR operations. Arithmetic instructions support operations like addition, subtraction, multiplication, and division. Bit-level instructions manipulate individual bits in registers and memory. Rotate and swap instructions rotate and rearrange bits and nibbles in the accumulator.