There were fan produced patches in the '80s and early '90s that had the Medical cross for nursing, one with Pi, and one with Lambda.
I'm sure other's have gone beyond that, but those are the ones I'm aware of that made it into some physical form.
The "Red Cross" for Medical is canon for at least nurses (it was created by Majel Barrett for her own use during TOS s2 & s3) and was occassionally used by extras filling in for her in the role. The "medical smock" worn by Bones in the KT also features a cross (in black rather than red) rather than the usual Sciences "globe".
The Pi and Lamdba logos are semi-canon at best, as they originally came from the chapter pages in the OS Tech Manual. They don't specify
what they are supposed to designate, but IMO from the context that one refered to diplomatic corps (it was in front of section on the Federation Charter et al) and the other was either Personnel or Training (it was in front of the "additional notes" section).
There are also models for Flight Ops (Sigma -
ENT: First Flight), Communications (satellite/radar dish -
TMP), Ground Forces (either the traditional "Command Star" (from MACO uniforms) or "four-pointed star" used on the UESPA flag and Franklin badge), Transporters (interrupted male stick figure, may also be holodecks/replicators), Sciences (eight-point atomic symbol from licensed clothing/accessories), Supply Corps (triangle or delta, from various props) and Engineering (eighteen-point star from licensed clothing/accessories, possibly for propulsion/power specialists given it's similarity to one of the Science symbols).