ok here goes:
They are about 4-5 suspension designs,
1. Single Pivot
This is the earliest full suspension linkage (before 4 bar and etc) it only has a single pivot, it moves in
an arc when the suspension compresses which caused a few problems such as pedal kickback and etc.
Famous Single Pivot designers: Cannondale and a few more which I forgotten

2. Single Pivot Linkages AKA Faux Bar
This bike has more linkages then single pivot but the rear wheel travel still moves in an arc, I believe
its slightly better then a single pivot linkage because they could reduce the arc travel compared to
single pivot but some riders disagree. Anyway single pivot linkage bike has a pivot on the seat stay
and not on the chain stay. They are a lot of famous Faux Bar bikes.
Famous SPL designers : Turner, Kona, Trek, Rocky Mountain and a few others which I forgot again

3. Horst Link AKA 4 bars/true 4 bars
Horst Link is owned by Specialized and they have licensed it to some companies like Nicolai, Titus
etc, Horst Link is one of the best linkage in MTB industry, it has eliminated most of the problems of
a single pivot or single pivot linkage bike and its rear wheel movement doesnt move in a arc.
Another famous 4 bar is Ellsworth ICT, orginally created by Dave Turner (Turner bikes) but was
patented by Tony Ellsworth, ICT is very similar to horst link but after a few slight tweaking, they
manage to make ICT into their own patent, if Im not mistaken the difference between ICT and HL is
HL uses a strut style shock linkage while ICT uses rocker style shock linkage. HL and ICT has a pivot
on the chain stay. Another noticable true 4 bar is Rocky Mountain ETSX, really interesting concept and
I suggest you to check on their website for more info bout their 4 bar, www.bikes.com
Famous 4 bar's: Ellsworth, Titus, Specialized, Nicolai, Rocky Mountain and others.

4. Virtual Pivot Bikes
Virtual Pivot is Mountain bike latest suspension linkage design, the advantage of this is the
manufacturer doesnt need to pay Specialized or Ellsworth for their linkage and Virtual pivot linkage
could be designed to have a different wheel path for each wheel travel progression, eg: from
sensative to firm as the travel increases. VPP bikes is mostly recognizable by having a bar on the
front portion of the bike rear triangle. Famous VPP tech is DW-Link, Santa Cruz's VPP, Giant's Maestro
Famous company that uses VPP : Ibis, Santa Cruz, Iron Horse, Giant, Felt and etc.

Those 4 is the most commonly and available on the market now, below is a summarize of older linkage that are not commonly used anymore:

1. Soft Tails
This bike doesnt have a pivot near the bottom bracket area, instead they replace it with a elastomer
or some sort to make it bump absorbtion, the linkage mostly are used for XC bikes because XC doesn
need too much travel, this linkage style died out because the travel is limited. Cannondale still make
soft tails.
2. Monolink
They are still a few companies that uses monolink, the reason why people dont really like monolink
is because as the suspension absorbs bumps, the BB will move with the swingarm (monolink rear
triangle is attached to the BB which connects to the crankset) making your leg bend more. If im not
mistaken, Mongoose, GT and Maverick uses monolink.

Well basically that is all the suspension linkage that are available in the market nowadays, they might be more but I cant remember any and I might make some mistakes so feel free to correct me and sorry for making this post uber long .