Depending on who I am talking to, I have found the shroud can be useful. Knowledge and the ability to discuss rationally and intelligently are critical when people have questions. Each person is unique and has different ways of understanding the world, so tailored witnessing/seed spreading, as needed, possible, and (most importantly) as directed by The Holy Ghost. Questions about the shroud are a giant billboard saying, "OPEN DOOR! OPEN DOOR! OPEN DOOR!" No matter where the person heard about the shroud, no matter the source's perspective, Jesus' Name is getting out to unbelievers.
From a fairly detailed and balanced article about the shroud (with references), albeit Orthodox:
From The Shroud of Turin: A Mystery Across the Ages (Fr Alexey Young), "As Orthodox Christians, do we need the relic of the Lord's Shroud? As far as the fullness of the Faith, "given once and for all to the saints," is concerned, we do not. The image on the Shroud adds nothing doctrinal to what has already been revealed; neither does it take anything away. Had it not survived Apostolic times, as some think, our faith in Christ and His Church, the Ark of Salvation, would be the same. Nor do we seek after signs and wonders to confirm our faith in Christ. On the other hand, the Shroud provides a visual document of something that the Evangelists describe in only a few terse words: "They crucified Him."
"Some years ago Archbishop Antony of Western America and San Francisco (Russian Church Abroad) said, in response to a question about the Shroud, that many Orthodox Christians, including some prominent bishops and clergy, accept the authenticity of the Shroud, "but," he added, "since the Turin relic does not touch upon the dogmatic teachings of the Church, no 'official' view is possible." Even the Roman Catholic Church remains reserved on this matter."
"A precise refutation of forgery charges is provided by a statement released to the press by the American experts at their final meeting:
The scientific consensus is that the image was produced by something which resulted in oxidation, dehydration and conjugation of the polysaccharide structure of the micro-fibrils of the linen itself. Such change can be duplicated in the laboratory by certain chemical and physical processes .... However, there are no chemical or physical methods which can account for the totality of the image, nor can any combination of physical, chemical, biological and medical circumstances explain the image adequately...
"We can conclude for now that the Shroud image is that of a real human form of a scourged, crucified man. It is not the product of an artist. The bloodstains are composed of hemoglobin and also give positive test for serum albumin.
"The image is an ongoing mystery, and until further chemical studies, perhaps by some scientists in the future, the problems remain unsolved."
"The image on the Shroud vividly tells us, in ways that words often cannot, what unutterable suffering was endured for our sake, and the high price with which cur souls were ransomed from eternal death.
Together with this universal significance which applies to all Christians at all times, the Shroud may also be said to be uniquely relevant to our 20th century, in which science has had such a powerful voice. Some believe that this image was encoded on the fibers of the cloth like a time capsule intended specifically for our materialistic age, when only the tools of modern science could begin to decode or unlock its secrets, when belief in God would be so weak or non-existent that even faith in science would testify to "the things of God."
There is a poster, plastered on walls in the Soviet Union, which shows a smiling astronaut flying through space. The caption reads: "There is no God," For individuals raised under the forced domination of 'scientific-atheism," the inability of scientists to disprove the Shroud does not go unnoticed. And there is reason to believe that the scientific evidence in favor of the Shroud's authenticity has been instrumental in opening doors to faith behind the Iron Curtain. (A report on the Shroud, written by a scientist in the Soviet Union, is said to be circulating there in Samizdat.)
We, too, in the free world, have been greatly influenced by the scientific-materialist outlook. And it seems that now, at a time which many believe to be the 11th hour, the suffering yet serene face looking at us from the Shroud confronts us with the REALITY of Jesus Christ. Can it be that in this age of diminishing faith, when even believers are crying out "Lord, help Thou my unbelief," the Lord in His mercy has condescended to reveal Himself to men in a special way, that seeing they might believe and exclaim with Thomas: "My Lord and my God!""
No link, as Orthodox website.