I have seen a few photos of stiff rigs and tied a few (badly) myself. Let's see yall's pics of stiff rigs and reverse combis. (yes, I know they are different)

I rarely use a stiff rig or reverse combi but I do use the hinged stiff rig.

That the one with the swivel? Stiff end on bottom of the swivel, braid on top, then swivel to the main line

yeah, kinda. I use a coated braid like snake bite and strip back the coating to show the soft braid under. I tie my hair rig to where the soft braid is exposed to less than an inch above the eye of the hook. I use a ringed swivel to allow the bait to be picked up from any direction yet still remain stiff towards the stripped back hinge even pushed up into the lead for the bolt effect.

Ted -- here you go -- this is the set up I caught the 47 on a few days back -

25lb Kryston Snake Bite, with a 1"+ hinge - as I wanted around a 2" pop up. Size 8 Nash Fang X hook, with about 15 turns on the knotless knot -- leaving the hair exiting the shank pretty much opposite the hook point. Split shot just anchors the 10mm long piece of Zig foam - but could be substituted with tungsten putty.
The loop is a figure 8 overhand - and the sleeve on there is a helicopter sleeve - that slips up onto the Korda Kwik Klip on the leader swivel.(Very simple quick change hooklink)

You were also asking about running rigs - so, that is a 36" Score heavy leadcore leader, with a size 8 barrel swivel spliced to it, a rubber shock bead, running lead ring with a small Korda Kwik Klip attached to a Fox lead.
Pretty simple stuff -- and not that difficult to build - if you have all the constituent parts!! A very deadly rig - when presented correctly
I hope this helps mate!!

Scott/Brid, thanks for yall's help.

Scott, you taught me that rig at LBL and that is how I tie my many hair rigs. Super easy and works great.

Brid, I would love to learn these, and will.

Ted -- the braid coating is a pain to get off easily/accurately - so, I have a Korda Strippa tool, which makes it a piece of cake. A handy bit of tackle to have!!
After you finish tying the rig - it needs straightening. A good way is to hang it from a nail by the loop, with a 2oz lead hanging on the hook - then carefully heat it all up with a heat gun (or hair dryer)on low heat. When it cools - it will stay straight.

Fox cortex is a good hook link material too. I use it straight off the spool to make my rigs. Gotta say it's been good to be.

Here's a very informative letter from Dave Chilton from Kryston - pulled from a UK forum.
Top drawer reply from Dave Chilton at Kryston -

Hello Bobby,
If colour is the only criteria and in my opinion it should not be, then
either Mantis Gold in 15lb or 25lb or Quick-Silver Gold in 25lb or
Quick-Silver in 25lb would be the nearest match.
Many anglers do not realize that Carp have extremely poor eyesight when
compared to many species of fish.
Carp have evolved as bottom feeders through evolution. Like eels cats tench
and many other bottom feeders their sense of smell and taste is extremely
acute due to their highly efficient receptors and sensors but.....their
sight is extremely poor!
When compared to species such as trout who rely wholly upon vision for food
the carp can be considered almost blind.
All species who feed from the bottom have their eyes situated in a far
different place than trout, Rudd, barracuda and so on who rely on sight to
fill their belly.
If you doubt that carp have poor eyesight the next time we have sunny
weather simply throw some mixers on the surface and watch how many times it
takes a fish to get the mixer in its mouth. Many times the carp has to make
numerous attempts to swallow the mixer.
It knows the mixer is there because of the smell but it cannot see it easily
and the closer it gets to the food the more blurred the vision becomes
simply because their eyes are in the wrong place.
If a carp could see a lead weight camouflaged or plain it would never pick
up a bait so close to the lead. Carp learn by recognition, every time a carp
is hooked what is the first thing it sees? Yes a lead weight dancing in
front of its face for the duration of the fight.
If carp could see the lead weight in close proximity to the bait we would
never get a bite. Likewise the colour of the hooklength is far less
important than anglers think it is.
What is most important is not the colour of the hooklength but the manner it
affects behaviour of the bait.
Think about this. As the fish approaches your hookbait it has already
consumed numerous free offerings lets say fifty baits.
All of those freebies have all very easily entered the carps mouth with no
hindrances whatsoever because they are not unattached to line. As soon as
the carp tries to suck your hookbait into its mouth it behaves differently
because of all the ironmongery attached to it. This is why the most
important aspect of presentation is the behaviour of your hookbait and not
the colour of the hooklength.
Colour and camouflage will always be a secondary issue. This is why anglers
such as Simon Crow who have limited time will use nothing else than a super
supple hooklength. Or the other way is to set a trap with a heavy lead ultra
short hooklength (some use as little as two inches) and just sit and wait.
This last method undoubtedly leads to many rejections that the angler is
completely unaware of, if he sits there long enough eventually after many
hours or days a fish will make a mistake. The angler hits the run convinced
the rig has paid off again whilst being totally unaware of the number of
refusals he has had.
There is no best hooklength, all waters are different, take into account
snags, sharp bars, tree roots, heavy weedbeds, silt, sand plus dozens of
other variables and then make your decision.
You will get more mileage by alternating the length of your hair than making
wholesale rig changes, try one rod with a short hair, one medium and one
long or extra long. One rig that worked last week may not work this week its
all about what they want on the day and as conditions and temperature
changes so do the carps behaviour. By fishing three different types of hair
length you are not putting all your eggs in one basket.
When I used to fish on Tim Paisleys Mangrove water I always fished with
three different hair lengths, I was very successful on the Swamp often
getting multiple catches at weekends. As soon as a pattern formed with the
successful hair length I switched the other two rods over to the winning
combination.
There is much much more, but I hope that this information may have given you
a little food for thought.
Best of luck with your angling, if you want any more info just mail me.
PS. If you already knew much of this then sorry for teaching my gran to suck
eggs!
Regards,
Dave Chilton.

I have not found, since being in the USA, that stiff rig, combi rigs and the like are no more productive than a good braid tied knotless knot style is all that is needed for carp (and buffs at that).

I do on occasions use coated hooklengths with an inch or two stripped back at the hook end for flexibility but really only do this for anti tangle rather than because they are needed to fool the carp.

My rule is always K.I.S.S. and don't go making rigs that catch the angler rather than more fish.........

+++1000 I could'nt agree with you more Keith, over here there isnt much need for fancy rigs when a hair rig will do just fine. There is so much extra bit of kit out there that is not needed, just keep it simple stupid and save your cash for some tackle you will really need.

Keith - good points!! The rig that I posted was the one I was using with the bait boat. Last thing I wanted was to be sending a boat load of bait out - with a rig that might tangle whilst sinking - and thus ruining the whole 'plot'. That was my 'sit on' rod - and the anti tangle characteristics of the Kryston Snake bite are good - so, my confidence level about the rig being nicely presented was pretty high.
Yes - KISS is a good rule of thumb

Not knocking your rig Brid just imparting my POV