What was the most likely method of crucifixion that was used on Jesus?

If you believe the Holy Shroud of Turin to be the shroud which covered Our Lord when He was placed in the tomb, and that the image imprinted on it is really of His body, then it should provide some important clues. There is a documentary/exhibit called The Man of the Shroud which analyses the image in the Shroud in some detail, which you could look up.

Two relevant details:

• First, the Man in the Shroud has wounds in His forearms, just below the wrists, as well as the feet, and that one leg is bent more than the other; this would lend credence to the fact that He was nailed to the cross using one nail through each wrist and one nailing the centre of both feet together.
• The second are the wounds on His head, which show that He was wounded by thorns probably fashioned in a crown with a closed, hemispherical shape, as opposed to the ring-like diadem which is usually depicted in crucifixes. This is consistent with Ancient Near Eastern practices where crowns were usually closed; circlet-like crowns only started showing up in the West, after the Germanic invasions of the Western Empire.

Regarding the shape of the cross, the Shroud also supports a T-like or cross-like shape, though there are studies refuting the standard asphyxiation theory. According to those, Jesus would have been able to breathe more or less normally on the cross, and died of trauma and blood loss derived from His flogging.

Another controversy brewing in the Hebrew Roots Movement is the cross or stake debate. Here is a good article exploring this whole discussion, which includes some references to the “mark”.

Used with permission by the author.

Cross or Stake?

Good Luck in your search. This is a very contentious subject

In the absence of a quantitative model, likelihood is likely abused. If a matter such as this, which is the topic of an unsettled scholarly discussion, the only reasonable answer is that we don't know, Since there is no conclusive evidence, available and many of the available sources are contradictory and open for interpretation (see also here for a mix of perspectives).

Not intending to promote any particular view but just for the sake of the argument: In the above answer by bradimus writes: "The author of Barnabas clearly thought the cross was shaped like the Greek letter tau, T." which is later interpreted to be contradictory by him with Matthew 27:37, favouring the traditional cross shape. However hands can reach higher than the head, and the sign could still be above the head also in a T-shaped cross.

The story with with Moses and his outstretched hands, can also be interpreted in various ways. An alternative interpretation is that he keeps his hands as high as possible since that seems to make the difference to win the battle. According to Exodus 17:12, he sat down on the stone while they kept his hands UP.

That Christ's hands were nailed to the cross is firmly established by Scripture. Following the Resurrection, the Apostle Thomas doubted that Christ was alive:

So he [Thomas] said to them, "Unless I see in His hands the print of the nails, and put my finger into the print of the nails, and put my hand into His side, I will not believe." (John 20:25, NKJV)

When Christ appeared to Thomas, he instructed

Reach your finger here, and look at My hands; and reach your hand here, and put it into My side. Do not be unbelieving, but believing. (John 20:27, NKJV)

While this does not rule out that His arms were also bound, it does require that they were nailed.

The Romans used a variety of different shaped crosses to execute criminals. For example, Seneca the Younger, a contemporary of Christ, wrote

I see before me crosses not all alike, but differently made by different peoples: some hang a man head downwards, some force a stick upwards through his groin, some stretch out his arms on a forked gibbet. (Moral Essays 6.20)

Josephus, who lived just after Christ, records

the soldiers, out of the wrath and hatred they bore the Jews, nailed those they caught, one after one way, and another after another, to the crosses (The Wars of the Jews 5.11.1)

So it seems that they had a variety of choice and applied them as they saw fit. The typical crosses were |-shaped, T-shaped, +-shaped, and X-shaped. Scripture does not clearly state which shape was used, but Christian writings in the first few centuries of the Church already favored either T or + shaped. The Epistle of Barnabas (c AD 90-125) records

For [the Scripture] saith, “And Abraham circumcised ten, and eight, and three hundred men of his household.” What, then, was the knowledge given to him in this? Learn the eighteen first, and then the three hundred. The ten and the eight are thus denoted—Ten by Ι, and Eight by Η. You have [the initials of the, name of] Jesus. And because the cross was to express the grace [of our redemption] by the letter Τ, he says also, "Three Hundred." He signifies, therefore, Jesus by two letters, and the cross by one. (Barnabas 9:7)

In Greek numerals, 300 is written with a tau, resembling the Latin T. The author of Barnabas clearly thought the cross was shaped like the Greek letter tau, T. In Greek, Jesus is IHSOYS -- I and H being the initial letters.

A bit further in the epistle, he writes

Here again you have an intimation concerning the cross, and Him who should be crucified. Yet again He speaks of this in Moses, when Israel was attacked by strangers. And that He might remind them, when assailed, that it was on account of their sins they were delivered to death, the Spirit speaks to the heart of Moses, that he should make a figure of the cross, and of Him about to suffer thereon; for unless they put their trust in Him, they shall be overcome for ever. Moses therefore placed one weapon above another in the midst of the hill, and standing upon it, so as to be higher than all the people, he stretched forth his hands, and thus again Israel acquired the mastery. (Barnabas 12:2)

Thus Moses, with his arms outstretched is described as a type of Christ on the cross. This suggests a T- or +-shaped cross.

A little later (c AD 150), Justin Martyr wrote

God does not permit the lamb of the passover to be sacrificed in any other place than where His name was named; knowing that the days will come, after the suffering of Christ, when even the place in Jerusalem shall be given over to your enemies, and all the offerings, in short, shall cease; and that lamb which was commanded to be wholly roasted was a symbol of the suffering of the cross which Christ would undergo. For the lamb, which is roasted, is roasted and dressed up in the form of the cross. For one spit is transfixed right through from the lower parts up to the head, and one across the back, to which are attached the legs of the lamb. (Dialogue with Trypho, Chapter XL)

This seems to match with a +-shaped cross.

So it seems that the earliest Christians believed Christ was nailed to a T- or +-shaped cross rather than a |- or X-shaped cross.

If we consider what Matthew recorded,

And they put up over His head the accusation written against Him: THIS IS JESUS THE KING OF THE JEWS. (Matthew 27:37, NKJV)

which seems to rule out a T-shaped cross, the balance of the evidence favors the traditional Christian shape of the cross.