Do SSDs get fragmented, and if they do, is that an issue?

Yes, SSDs do get fragmented. Does it impact performance as much as regular hard drives? No.

Fragmentation just refers to the placement of files out of order. It's necessary on all storage devices without having to reshuffle all the data that has ever been written every time you write something.

Solid-state drives DO NOT require defragmentation. It may decrease the lifespan of the drive. Wear leveling technology purposely "fragments" the data to ensure the consistent life of the drive.

Source: OCZ

“Suffer”? No. Experience? Yes.

Fragmentation is simply when files are written to non-contiguous blocks. This is not a problem with a fresh, clean drive, but after a while, as the drive fills up and files are deleted, new files eventually begin to get written into whatever blocks are available which may not always be big enough for the whole file. There’s really no way around it (short of writing everything once to an empty drive then not writing anything ever again), not even with a better file-system.

However, as sblair pointed out, it’s not actually a problem with SSDs like it is with HDs because there is no head to physically move around the disk to collect data, so there is no performance penalty.

Also, as Marcin and Molly explained, SSDs need to scatter data throughout the whole drive to prevent the beginning of it from getting worn out while the rest of it remains unused. As a result, SSDs purposely fragment data to spread it around the whole drive. Also, you don’t want to defragment SDDs because not only does it defeat the purpose of spreading the data around, but all the extra writes wear it out faster. Robers gave a good explanation about how SSDs are different from traditional spinning disks and that the sectors on an SSD do not reflect the physical layout.

This is yet another benefit of SSDs over spinning HDs: fragmentation is no longer an issue.

SSDs : 5 (less power, less heat, faster, no fragmentation, smaller)
HDs  : 1 (longer lasting)

Caveat:

All this does not however mean that SSDs are the ideal, care-free storage solution. Aside from the fact that they wear out, there is a critical issue to be aware of. While it’s true that SSDs employ wear-leveling and their firmware manages sector mapping, that does not negate the fact that SSDs, like all storage devices that allow modification, do become fragmented which is death when you need to recover lost files.

Using a more tolerant filesystem like NTFS instead of FAT32 helps to some degree, but the fact is that a fragmented file is much harder to recover (if not flat out impossible) than one that is not fragmented. Of course in this case, “fragmented” means from the OS’ point of view irrespective of the physical layout.