Super Xfer by David Bolt & Greg Pfountz is a single drive file copier with three modes. The normal mode allows for copying of 78 files or 221 blocks at a time. Max Buffer mode allows for 233 blocks but only 6 files at a time and Max Files mode allows for 191 block and 258 files.

Found this older version of Unicopy written by the famous Jim Butterfield.  This version has support for tape drives which I believe is missing in newer versions.

Written by one of Commodore's all time greats, Jim Butterfield. Unicopy is a file copier that not only takes into consideration whether you have a wedge, monitor or linker in memory, it can also copy from disk to tape.

File Copier by Marty Franz and Joe Peter (1987 RSD) - A while back I posted this utility, but it was a hijacked version by some group called TSS/Tango Soft Systems (link).  Now that I've found what is probably the appropriate unscathed version, it's time to give the authors credit.  One thing I didn't catch before is that this copier has support for Warp-DOS which is the same as the 1571 Burst Speed, so use that if you've got the right hardware.  Additionally, there is a nice BAM viewer / editor, and a couple other creature comforts.

VFcopy by Marty Franz and Joe Peter (1987) is an excellent file copying program for the Commodore 64 / 128.  Besides being multi drive / device capable, it also has a built in Warp-DOS mode that is really nice for speeding things up.

X15 + Xfer v2.0 by Wild Ron has one of the strangest names I've seen for a file copier, but don't let that fool you, this is a really great program.  Not only is this a file copier, it handles both single and dual drive copying with a HUGE buffer for those using one drive.