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In the cartoons, I've seen Spider-Man shoot a variety of webs from his web-shooters. The ones I can name now are:
If these all come from the same device, how is he able to choose which type of webbing is shot?
(I'm as interested in the comics as the cartoons, FWIW.)
Marvel's official website has this diagram of Spider-Man's web shooters:
The Wikipedia entry on Spider-Man's webbing states simply that Spider-Man "can do many things with his webs" and lists many different types of webbing (the ones you listed included). All of these different types of webbing come from the same shooters; they just occur when the wrist devices are used differently.
To paraphrase the Marvel Wiki (emphasis mine):
The wristlets had sharp steel nipples which pierced the bronze caps when the cartridges were tightly wedged into their positions. The hand-wound solenoid needle valve was actuated by a palm switch that was protected by a band of spring steel which required a 65 pound pressure to trigger... The effect of the very small turbine pump vanes was to compress (shear) the web fluid and then force it, under pressure, through the spinneret holes which cold-drew it (stretches it: the process wherein nylon gains a four-fold increase in tensile strength), then extrudes it through the air where it solidified. As the web fluid exited the spinneret holes, it was attracted to itself electrostatically and thus could form complex shapes. The spinneret holes had three sets of adjustable, staggered openings around the turbine which permitted a single web line, a more complex, spun web line, and a thick stream... The 300 p.s.i. pressure in each cartridge was sufficient to force a stream of the complex web pattern an estimated 60 feet (goes significantly further if shot in a ballistic parabolic arc).
As Jeff points out over in this question, this has been discussed to death over on the Marvel Wiki... But, to save you wading through pages of debate:
Take a look at the Web-Shooters page on the aforementioned Wiki; it basically says what he has said a few times in the comics; they have settings that allow him to adjust what he shoots.
The effect of the very small turbine pump vanes is to compress (shear) the web fluid and then force it, under pressure, through the spinneret holes which cold-draws it (stretches it: the process wherein nylon gains a four-fold increase in tensile strength), then extrudes it through the air where it solidifies. As the web fluid exits the spinneret holes, it is attracted to itself electrostatically and thus can form complex shapes. The spinneret holes have three sets of adjustable, staggered openings around the turbine which permit a single web line, a more complex, spun web line, and a thick stream.
(Emphasis added to specifically address your question.)
To link them up to your specific question details:
In an info insert in an early Spider-Man issue, it's explained that he can change the setting by manually spinning the nozzle to different settings.
I can't find a source at the moment, but I recall the system changing to the nozzles spinning based on the pattern Peter taps out on the trigger.
In Amazing Spider-Man Annual #1 (1964), it was stated that Spidey could expel his webbing in three different ways by adjusting the nozzles on his web-shooters.
By adjusting the nozzle of his web-shooter in one easy motion, Spidey can eject his web fluid in any one of three different ways...
As a thin, incredibly strong line...
As a fine, quick-spreading spray...
Or as a thick, tremendously adhesive liquid...!
By Amazing Spider-Man Annual #4 (1967), he'd apparently modified his web-shooters so that he could vary the output based on how many times, and for how long, he depressed the triggers on his palms.
The amazing web-fluid can gush out in many different forms, depending upon how long Spidey's finger remains upon the sensitive control button...
A short tap releases a thin, cable-like strand, recommended for swinging... or stringing guitars!
A longer period of pressure releases more fluid, forming an icky, echhy, blochhy, sticky blob... useful in pasting a foe against a wall... like!
A series of brisk, staccato taps releases multiple cable shapes which can be formed into a decorative netting pattern!
A similar explanation was given in Spider-Man: The Ultimate Guide (2007).
TWO TAPS
Peter designed his web-shooters so that he wouldn't accidentally fire them every time he made a fist. Resting in the palm of his hand, the trigger works just like a computer's mouse. He must tap twice in rapid succession to release his webbing.
DIFFERENT STROKES
Peter has improved upon his initial design. He now switches between different forms of webbing by the way he taps his trigger. With a short second tap he releases a thin cablelike strand that is perfect for web-swinging. A longer second tap increases the strand's thickness for additional support. If Spidey prolongs the pressure on the trigger, web fluid squirts out in the form of an adhesive liquid that can paste a foe against a wall. A series of brisk taps discharges many thin strands that form a fine spray of webbing, perfect for blinding an opponent.
Spider-Man's entry in the Official Handbook of the Marvel Universe A-Z #11 (2009) provides detailed diagrams of both the mechanical and organic web-shooters, and states the following:
The effect of the very small turbine pump vanes is to compress (shear) the web fluid and then force it, under pressure, through the spinneret holes, which cold-draws it (stretches it: the process wherein nylon gains a four-fold increase in tensile strength), then extrudes it through the air where it solidifies. As the web fluid exits the spinneret holes, it is attracted to itself electrostatically and thus can form complex shapes. The spinneret holes have three sets of adjustable, staggered openings around the turbine that permit a single webline, a more complex, spun webline, and a thick stream.
Good question, but when it comes down to it, the shape, size, viscosity, and adhesiveness of Spider-Man's webbing is determined by ... the mystery of Peter Parker's ingenious design! It's kind of like the enigmatic spider sense - it just works that way because the writer of the story wanted it to. It's great to think logically, but asking a question like this about a character who is only partially believable is over-thinking it a bit. Although apparently, Spider-Man's webshooters have sensitive trigger buttons that require variations of palm-squeezing techniques (using alternate fingers or tapping the trigger twice, for example) to fire different forms of webbing. Exactly what these little slight-of-hand tricks are is pretty much unexplained, or at least, vary from comic to comic.
