In Illinois ARCH typically uses IREACH (155.055) on vhf to contact and coordinate landing with fire/ems units on scene. There are several IREACH to/from STARCOM21 patches but most fire departments use a vhf radio because the patches are not always reliable. If you listen to IREACH you will hear ARCH a lot but won't hear the on scene units. This is because the helo is at altitude and obviously will have better propagation.
In Missouri, specifically the St. Louis Area, they use VFIRE21 (154.280) I believe.
ARCH itself has its own UHF frequencies it uses to communicate with it's dispatch.
ARCH to hospitals, they will probably use whatever VHF or UHF (MED channel) the hospital uses for inbound patient reports. MERCI 340 (155.340) is a good place to start
Scanner Frequencies and Radio Frequency Reference for Hospitals (St. Louis County)
www.radioreference.com
Scanner Frequencies and Radio Frequency Reference for EMS / MedEvac / Public Health (Illinois)
www.radioreference.com
Scanner Frequencies and Radio Frequency Reference for St. Clair County, Illinois (IL)
www.radioreference.com
I can tell you that if you're trying to monitor EMS to hospital comms you will find that it's pretty much non-standard these days as everyone does it a bit differently. You might hear stuff on 155.340 one day and then nothing. This is because if I'm in the back of an ambulance and need to give an inbound report I can either call on cellphone, call on vhf (MERCI channels), call on UHF (MED Channels), or a hospital talkgroup if it has one. Sometimes it gets even more confusing because some areas will patch a new P25 talkgroup into a VHF or UHF hospital channel.
Every hospital has radios but to what capacity is unknown. If they utilize a MERCI or MED channel, they usually have there own hospital specific PL tone so they don't have to hear reports to other hospitals all the time (linked above). I know most in IL have at least 1 starcom radio, but again what talk groups, it depends.
Like I said earlier, cell phone patient care reports are the norm in St. Louis. In IL it's very department/hospital dependent.