What priority if any is given to Emergency Services in flight clearance?

I work as a fixed-wing pilot for a hospital-based Air Ambulance program. Let me add my experience to the ATC orders accurately cited by Pondlife's answer: the MEDEVAC call sign (the old "Lifeguard" is still used when filing the flight plan) indicates a need for priority handling.

ATC is often helpful in providing us with priority routing. This includes giving us direct clearances, moving us up in the landing que, allowing us an unrestricted approach speed, and otherwise expediting our flight. However, ATC does not always provide priority handling, and I have often experienced ground delays of 5-10+ minutes as ATC apparently prioritizes other traffic, including air cargo and the like. This is rarely a big problem as patients are typically stable, but it can be frustrating when priority handling is expected.

However, if medical staff determine that a patient is in critical condition, we will announce this to ATC—typically accompanied by a request for a specific clearance or change to a previous clearance. This announcement re-iterates our need for priority handling and indicates the urgency of the situation. In this way we can work with ATC to best care for the patient. We will often make such an announcement when checking on with approach control for their information with no specific accompanying request. Occasionally, however, we will need to request such priority handling because ATC has issued us speed restrictions or delaying vectors to put us behind other traffic. ATC is very good at working with us when we do need the priority handling.

In other words, ATC usually automatically gives us priority handling, but when they don't—and we need it—we ask for it and they are very good at working with us to provide the priority handling.

In my experience, yes they will hold for emergency related vehicles. I fly out of KPNE often, where a lot of the Philly Police and Emergency choppers are kept. I have been held for their takeoff before. I don't know if this is regulation or simple courtesy but I can assure you it happens.

On a similar note, I once spoke to the designer of the Philly Bravo airspace about this very issue. He explained that when the airspace was redesigned (a few decades ago now) the shelves were deliberately placed so that all of the hospital helipads were accessible without entering the bravo. Basically the emergency choppers can come and go with out clogging ATC for clearance in and out of the Bravo.

In uncontrolled airspace this is not really an issue and emergency helicopters will rarely go into class A. So this leaves you with Class B, C and D departures and arrivals. In the case of Philly the helicopters are stored at a less busy airport outside of the bravo for operations commonly under the shelf. This most likely varies heavily from place to place and their may be arranged understandings in different Bravo's.

(I'm assuming you're asking about the US, based on your profile.)

The FAA's ATC orders section 2-1-4 has a list of aircraft operations that should be given priority, including emergency service flights, but an aircraft in distress always has the highest priority:

Provide air traffic control service to aircraft on a “first come, first served” basis as circumstances permit, except the following:

[...]

a. An aircraft in distress has the right of way over all other air traffic

[...]

b. Provide priority to civilian air ambulance flights (call sign “MEDEVAC”). Use of the MEDEVAC call sign indicates that operational priority is requested. When verbally requested, provide priority to AIR EVAC, HOSP, and scheduled air carrier/air taxi flights. Assist the pilots of MEDEVAC, AIR EVAC, and HOSP aircraft to avoid areas of significant weather and turbulent conditions. When requested by a pilot, provide notifications to expedite ground handling of patients, vital organs, or urgently needed medical materials.

Other flights that get priority are presidential ones, search and rescue etc.