OK so time for a little zoology catchup for those readers not immediately familiar with these terms.
A spiracle is a hole in the head of some groups of jawed vertebrates that opens behind the eye and leads to the chamber behind the mouth (we would normally call this the 'throat' but for us zoologists it is specifically called the pharynx). It is an ancient feature that was almost certainly present in the common ancestor of living jawed vertebrates but has been secondarily closed over in the vast majority of living vertebrates (us included). Sharks are probably the most familiar living vertebrates that retain a spiracle (although some sharks lose theirs as well).
A zebra shark with a close-up of the eye and spiracle. The spiracle is arrowed.
Bichirs are a group of nine species of really weird freshwater fish from Africa that are classified in the family Polypteridae. They are part of the enormous ray-finned fish group that includes such familiar and diverse fish as salmon, seahorses and sunfish (and many more besides). However they are an extremely early branch and are the sister group of all other living ray-finned fish. Among the oddities that bichirs display is the presence of a pair of lungs, and they are well known for supplementing their oxygen intake from their gills with air breathing.
A bichir (Polypterus weeksii). Image from Wikimedia commons.
As you had probably guessed bichirs also retain a spiracle. In this case they open on the dorsal surface of the head, although they are not immediately obvious in pictures because they are covered by flap-like bony valves. The question is now what do bichirs do with their spiracles? It had been suggested along time ago that they breathed air through them but this suggestion became more-or-less forgotten. It took a team of scientists applying all sorts of techniques, such as CT scanning, behavioural cinematography and air pressure measurements to demonstrate conclusively that they do indeed breathe through their spiracles. Indeed the great majority of all breaths are taken through the spiracles This makes a lot of sense, gulping air for a fish is energetic and immediately draws attention to your location, whereas quietly protruding just the top of your head to take some quiet lungfulls through your spiracles seems like a really good idea.
Enter stem tetrapods. What is a 'stem' tetrapod. 'Stem' is a term used in cladistics classifications in conjunction with the term 'crown'. A crown is a clade consisting of all living members of the clade, the most recent common ancestor of all of those living members, and all descendants of that common ancestor, whether or not they are extinct. For example crown clade tetrapods include the common ancestor of all living mammals, birds, reptiles (not actually a natural group - unless one includes birds) and amphibians and all descendants of that ancestor. By this definition Tyrannosaurus is definitely a crown clade tetrapod even though it is most definitely extinct. However other more ancient 'tetrapods' like Acanthostega are not crown-clade tetrapods because it diverged away from the tetrapod lineage before the origin of the common ancestor of all living forms (as can be determined from its anatomy which retains some decidedly 'fishy' traits not seen in any living tetrapod). This diagram should clarify these points.
So back to those stem tetrapods. The skulls of these guys show a hole (or open notch in more derived taxa that have lost some of the bones that encase the pharynx region of the head) in exactly the same position as the spiracles of the bichirs. It seems very likely that spiracles were retained in the early tetrapod lineage as well. This is all well and good. It is apparent that the early stem tetrapods like Acanthostega were largely aquatic creatures, and that if it weren't for their relationship to later tetrapods would be just regarded as another unusual group of 'fish'.
Skull and partial skeleton of Tiktaalik roseae a rather 'fishy' stem tetrapod. This is one that retained fins, instead of limbs with digits and so would fall on the non-tetrapod side of an arbitrary divide in more conventional classifications. Note the large obvious spiracles.
A model of the skeleton of Acanthostega gunnari a more 'tetrapody' member of the tetrapod stem group. This guy does have limbs with digits instead of fins and so is usually classified as a tetrapod despite retaining such fishy features as an internal gill chamber in adults, a fishy tail fin and, of course, spiracles. Both images from wikimedia commons.
But how far up the tetrapod lineage did spiracles persist? One could argue that no living tetrapod, be it frog, caecilian, snake or whale has a spiracle and therefore that common ancestor of all of these (i.e. the common ancestor of the crown group) similarly lacked a spiracle.
But not so fast! Convergence is a real feature of evolution, and seems to happen particularly easily when one is looking at the loss of characters, in this case the loss of spiracles. Temnospondyls are a group of early tetrapods that appear to be related to modern amphibians (frogs, salamanders and caecilians), and, if so, would be part of the tetrapod crown. I freely admit that this is by no means settled, and their is also evidence that temnospondyls are stem tetraopods outside the tetrapod crown (indeed this is the position I took when I wrote my big paper on temnospondyl relationships some 15 years ago - oh how time flies). However for now I accept that temnspondyls are indeed stem-amphibians with small dissorophoid temnospondyls like Gerobatrachus being particularly close to modern amphibians, in effect they are transitional forms.
Temnospondyls also have dorsally-facing open notches at the back of the skull between the roof of the braincase and the cheek region, exactly the same spot as they occur in Acanthostega. These notches become partly enclosed by outgrowths of the surrounding bones in several different groups of temnospondyls.
Kupferzellia (or Tatrasuchus) wildi, a Triassic, German temnospondyl with almost completely enclosed spiracular notches at the back of the skull. Image from wikimedia commons.
In the past these notches have been called 'otic notches', with 'otic' referring to the ear. It was envisioned that the notch was the frame across which the ear drum was stretched. The idea received some support from the orientation of the stapes in temnospondyls. The stapes is the ear ossicle that attaches to the ear drum and transmits these vibrations to the middle ear (via another two ossicles in the case of mammals like us humans). In temnospondyls the stapes forms a rod that sweeps up and out from the ear region of the braincase towards the otic notch.
However there are a few observations that would hint that the otic notch was not a frame for an ear drum. Firstly the stapes, although rod-like for most of its length, is usually quite large and chunky and not apparently well suited to transmitting delicate sound vibrations. Secondly the end that fits into the ear region of the braincase does not always maintain a mobile joint. In adult Mastodonsaurus the stapes rigidly sutures to the braincase thus immobilising the stapes. Not a good move when a freely mobile ossicle is optimal for transmitting sound. Thirdly, as noted by Warren and Schroeder (1995) in a particularly well-preserved and uncushed temnspondyl skull the tip of the stapes does not actually protrude precisely into the otic notch as one would expect if it was attached to the inner side of an ear drum. Rather the stapes seems to run along side of the chamber below the otic notch so that in life it would have been embedded in the wall of the chamber. Consequently Warren and Schroeder proposed that the stapes formed structural support for a persistent spiracle. Given the similarity between the so-called otic notches of temnspondyls and the probable spiracles of stem tetrapods, I'm inclined to agree.
I would add one more observation in support of persistant spiracles and spiracular breathing in temnospondyls. There is no doubt that the majority of temnospondyls were semiaquatic to fully aquatic. A look at the heads of modern aquatic tetrapods will all show dorsally migrated nostrils that are often mounted on low prominences so they breathe without having to lift the head out of the water. A look at the position of the nostrils of temnospondyls does not show this feature. In most stereospondyls (the subgroup of temnspondyls apparently particularly well-adapted to an aquatic existence usually have laterally located nostrils that are low down on the snout, usually separated from the gum-line of the upper jaw by a narrow bar of bone. This doesn't make much sense, unless of course they weren't breathing through their nostrils at all but were relying on their appropriately placed spiracles to get a breath of air.
A comparison between a nostril breathing crocodile (left) and a potentially spiracular breathing temnospondyl (Broomistega putterilli) on the right. Arrows indicate the position of the nostrils. Note the poor position of Broomistega's nostrils for breathing while in the water and the spiracular notches at the back of the skull. As an aside the Broomistega skull was rapid prototyped from CT data gathered from a scan of a specimen embedded deep in a Thrinaxodon burrow cast, no-one has ever actually seen the fossil itself. To be honest this 3D print is probably far better than any attempt to mechanically prepare the skull out of its burrow cast. Photo by Adam Yates.
The up shot of this is that it is quite likely that the most recent common ancestor of all living tetrapods, the direct ancestor of crown group tetrapods, was infact a spiracular breather. Now that is something I don't think anyone has proposed before.