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• dash
• the sh of NetBSD (also based on the Almquist shell).
• The sh of FreeBSD 9.2 or older (see the change in 9.3)
• yash
• zsh with versions prior to 5.1 in ksh or sh emulation (or for export var="$(cmd)" where zsh would perform word splitting otherwise (not globbing)).

• bash
• ksh (all implementations)
• the sh of FreeBSD 9.3 or newer
• busybox' ash-based sh (since 2005)
• zsh

If the export/local command name itself is quoted (even in part like "export" a="$b", 'ex'port a="$b", \export a="$b", or even ""export a="$b"), the quotes around $b are needed except in AT&T ksh and mksh.

If export/local or some part of it is the result of some expansion (like in cmd=export; "$cmd" a="$b" or even export$(:) a="$b") or in things like dryrun=; $dryrun export a="$b"), then the quotes are needed in every shell.

For command export a="$b", the quotes are needed in every shell but mksh, ksh93 and bash -o posix (with the same caveats about command and export not being the result of some expansion).

• dash versions 0.3.8-15 to 0.5.10.2 (see change).
• the sh of NetBSD (also based on the Almquist shell).
• The sh of FreeBSD 9.2 or older (see the change in 9.3)
• yash
• zsh with versions prior to 5.1 in ksh or sh emulation (or for export var="$(cmd)" where zsh would perform word splitting otherwise (not globbing)).

• bash
• ksh (all implementations)
• the sh of FreeBSD 9.3 or newer
• busybox' ash-based sh (since 2005)
• zsh
• dash 0.5.11 or newer.

If the export/local command name itself is quoted (even in part like "export" a="$b", 'ex'port a="$b", \export a="$b", or even ""export a="$b"), the quotes around $b are needed except in AT&T ksh, mksh and recent versions of dash.

If export/local or some part of it is the result of some expansion (like in cmd=export; "$cmd" a="$b" or even export$(:) a="$b") or in things like dryrun=; $dryrun export a="$b"), then the quotes are needed except in recent versions of dash.

For command export a="$b", the quotes are needed in every shell but mksh, ksh93, recent dash, and bash -o posix (with the same caveats about command and export not being the result of some expansion in shells other than dash).

For command export a="$b", the quotes are needed in every shell but mksh and ksh93 (with the same caveats about command and export not being the result of some expansion).

For command export a="$b", the quotes are needed in every shell but mksh, ksh93 and bash -o posix (with the same caveats about command and export not being the result of some expansion).

My advise would be to always quote.

My advice would be to always quote.