Hundreds, depending on whether you include the ones that Linux supports in user mode (via FUSE), and ones that are used only in embedded computers.
The ones that Linux supports natively (can boot off of) are:
FAT, FAT16, FAT32, ext, ext2, ext3, ReiserFS, Reiser4, NFS, ISO9660, UDF, XFS, and JFS.
Ones that Linux distributions often support using FUSE:
NTFS, AFS, QNX6, Coherent, HFS, HFS+, RAMFS, Minix, HPFS, BFS, XIAFS, and UFS.
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Dozens. Linux can read almost every file system in existence, and can write to most of those. Linux can boot off of a FAT16, FAT32, ISO9660, ext2, ext3, ext4, JFS, XFS, ReiserFS, or Reiser4 partition.