To determine which statements regarding Solid State Drives (SSDs) are correct, we need to analyze each statement individually based on the properties and functionality of SSDs.

1. Statement 1: Spinning speed of SSD is zero.
   This statement is correct. SSDs do not have any moving parts, unlike Hard Disk Drives (HDDs) which have spinning disks. SSDs use flash memory to store data, which makes them faster and more reliable as there are no mechanical latencies associated with spinning disks.
2. Statement 2: SSD consumes more energy compared to hard disk drive (HDD).
   This statement is incorrect. Typically, SSDs consume less energy compared to traditional HDDs. This is because SSDs do not have any moving parts, whereas HDDs require energy to spin the disks.
3. Statement 3: SSD could be used as virtual memory when RAM is insufficient.
   This statement is correct. An SSD can be used to increase virtual memory when the system's RAM is insufficient. Virtual memory uses storage space on the drive to simulate additional RAM, helping improve system performance when physical RAM is maxed out.
4. Statement 4: SSD can replace the processor.
   This statement is incorrect. An SSD cannot replace a processor as they perform entirely different functions. The processor (CPU) executes instructions and processes data, whereas an SSD is a storage device that retains data.

Based on the analysis above, the correct statements are 1 and 3. Therefore, the correct answer is "1 and 3".