Well it depends on edition and what contacts your GM allows however I suggest the following types:
1. Bodyguard: Very useful for certain kinds of characters like "Deckers" and "Mages" that will be spending time 'out of it' with astral projection or jacked into the matrix. Being able to call your contact and hire him to watch your meat body "just in case" for prolonged assignments can be quite useful. SR 4 lists the going rate as 250 Nuyen a day for typical bodyguard fees. With a decent loyalty score and a bit of extra money, this can be a useful contact and easily 'covered' with your up front money most times.
2. Docwagon Paramedic: Another useful contract for any game where your regularly using a "Docwagon" Contract. Having a contact/friend who is willing to go that extra mile for you when you hit the button and will watch your badly injured body a bit more carefully than normal when someone might be out to finish you off can go a long way.
If you have other contacts (Street Doc, or something more skilled/relevent) this contact can also be used to set up a deal to have Docwagon drop you off with your other contact (and thus disappear) instead of the normal services. All told a nice contact to have handy.
3. Any kind of a "Cop" contact for the area your working in can be useful as a way of getting access to police files and evidence, not to mention being a potential "way out" in the unlikely event that your character gets dropped by an taser shot from a police officer, or winds up in a hospital after a "loud" situation that has the cops showing up shortly after the Ambulance. Such a contact can't perform miracles of course, but still it opens doors.
4. Any kind of "Gang" contact for the area you want to set up your 'lifestyle' in. Having a contact means you might actually get something for kicking some money over to the gang once in a while. This can be someone to watch your meat body, or just to reduce the chance of someone breaking in if say your supporting a "Low" Lifestyle yet are keeping a lot of tools and gear in your crappy apartment you'd prefer not to see walk off.
5. Any kind of "Decker" or "Hacker" type contact is good as it can let you hire an information modification specialist without having to rely on other PCs who probably have their own things to worry about (like taking the time to write programs) between adventures. Even if your character is a Decker or Technomancer it can be good to have someone to hire for "Side research" while your busy, or to help you develop programs.
6. A "Fence" is also good because part of Shadowrunning is to steal stuff, and since your character probably doesn't need 10 assault rifles or pistols, you can make a bit of extra money by selling some loot on the side. Encumberance means you probably can grab everything carried by the baddies during an adventure, but most GMs (especially if you have a contact for it) don't object to PCs making a little extra money on the side.
7. A good Assasin/Bounty Hunter/Mercenary contact can be useful too. Both for the same purpose of a Bodyguard (above) though usually more expensive (players don't want to waste time watching your meat body if they can be out doing something), and also because with good loyalty you not only giving yourself a weapon you can use, but preventing that guy from coming after you. I mean logically there are only so many super-deadly assasins in any given area.
Think of the thing from the recent "Ocean" move Preview where George Cloony tells the guy "I know all the guys you could send after me, and they like me better than you". It's also useful as a Shadowrunner if your not just the guy who has to worry about having assasins sent after you, but you know a few you can send after other people. While it's totally a GM call, it can be occasionally useful to be able to say "alright well, if this guy is going to be a pain, I'll call up my buddy Reaver the elite assasin and offer him 20k Nuyen to whack this guy first". If your supposed to be some kind of connected mini-fixer youself (like a Face) knowing a few Assasins/Mercenaries/etc... that you can call and hire, in addition to the PCs
is a big plus.
It shouldn't be your first choice as a contact, but it's a good one to have, but also one that is dependant on the GM since some (but not all) GMs are resistant to the idea of PCs removing obstacles by hiring their own assasins. Even in cases where the GM doesn't allow this to be practical, it's still a pretty good guy to be able to call when you need someone to watch your back during Astral Projection/VR Hacking during a run, and such characters are usually a bit tougher than a regular bodyguard (though the book rates are predictably higher).
What's more I've been in a few games where in really bad situations the PCs have basically thrown around a lot of their own money, and called in a lot of favors to establish a small "shadow army" of combat contacts. You know like when due to bad luck (or perhaps stupidity) the warehose HQ the PCs built is about to be blitzed by a corperate Strike Force or the combined forces of a couple of Triads or major gangs, and for whatever reason the PCs can't (or don't want) to move since it might cost them a lot of stuff like extra vehicles, workshops, libraries, etc... Some of those contacts with few informative uses but high listed combat statistics do come in handy from time to time.
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As far as not allowing Fixers and Mr. Johnsons, yeah I can understand that to an extent. I've run into a few GMs who have a hang up with that. The big reason is that especially with "buddy status" or "high loyalty" it can limit the GM's options a bit as far as betrayal, and them keeping a lot of information close to their chest. Since you paid for the contact he can't really have you kill your own "resource" and while nice to have these guys watching out for you from square 1 (and making a lot of sense, especially how a lot of runners operate in the novels) it does as I said, limit options. Though in a few cases GMs have felt it's a waste of a contact space since your going to have a Fixer or Mr. Johnson anyway, so why pay for it?
Though to be honest this is a double edged sword. Not allowing characters to have "safety" with these contacts does lead to a lot of institutional paranoia (so to speak) even when the GM doesn't intend it. Some of the worse frag ups I've seen in games have occured simply because of the way the GM ran the contacts and didn't let the PCs have their "hooks" into their employers or arrangers (so to speak) so thus had the PCs complicating things needlessly trying to second guess a betrayal that was never intended. While a certain degree of paranoia does belong in the setting, it can go too far, and even in Shadowrun there should be SOME Level of comfort/security especially for PCs. Oddly in many games that comes from working through a fixer that the PCs know has a huge loyalty or "buddy" rating.
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In general I tend to find corperate/tribal/foreign/crime contacts to be less useful. They are things to develop later in a character's career, but not something you should start with or spend a lot of time cultivating unless your building a character who for whatever reason has a ton of contacts.
The reason is that such contacts are only useful in very specific situations. If an adventure doesn't involve the Corperation (and there are a lot of them) that your Contact is from, then he's not likely to be very useful. The same can involve Tribes (which are only really useful if your not only leaving the metroplex but going to a specific area), or Organized Crime contacts (since if the specific syndicate you know isn't involved, it's not likely to help). Very specialized
contacts might add flavor, but are usually at the bottom of the list.
As brutal as it sounds, you can only have so many friends (to begin with) so each one that your 'picking' should be carefully considered based on what they can do for you, and whether it's practical to use them. A super-high end gunrunner isn't very useful to a starting Shadowrunner because you can't afford his prices. The more pieces you can put on the board 'now' so to speak increases your chance of being able to obtain more contacts to do other things later simply by surviving and getting more money/karma to throw around.