The Lutheran Quarterly has published several essays marking the 500th anniversary, one of which is Sources For and Against the Posting of the 95 Theses, by Volker Leppin and Timothy Wengert".
The principal source for the story of the nailing of the theses is in the preface to an edition of Luther's Latin works, penned by Philip Melanchthon shortly after Luther's death in 1546. In it Melanchthon, Luther's very close friend and confidant, says:
Luther, burning for the study of godliness and outraged by Tetzel's ungodly and nefarious sermons, published propositions concerning indulgences, which are extant in the first volume of his works, and publically posted them at the church which is next to Wittenberg Castle on the eve of the feast of All Saints in the year 1517.
Melanchthon was in the habit, when writing letters, of giving not just the date of writing but something special about that date in history or scripture (as assumed at the time). He refers, on 31st October, to the Posting of the Theses on the church as occurring on that date in letters he wrote in 1552, 1553, 1554, 1555, 1558 and 1559. (He died in 1560.)
It is clear Melanchthon regarded this as a significant and seminal event, worth commemorating, and that it happened on 31st October 1517.
However, Melanchthon did not arrive in Wittenberg until August 1518 and so could not have been an eye witness or involved directly at the time. Nevertheless he was close to Luther, and collaborated very much with him, and so he might be expected to know.
Separately there is a note by Georg Rorer, another close associate of Luther, who documented much of Luther's work, in the back of a book published some time after 1540, saying that on 31st October, 1517, the theses were published on the doors of the churches (note plurals).
Thee is also a letter written in the late 1540s by Georg Major, who was a choirboy at the Castle Church in 1517, and went on to become a professor himself, saying that he was an eyewitness of the event.
However, questions have been raised over whether the note really was by Rorer, and why the choirboy turned professor would have remembered such a relatively mundane event at all, since people pinning notices to the door was not unusual, and he could not then have known he was seeing a seminal moment in history which would so thrill future generations. It has been alleged he may have been influenced by Melanchthon's account.
Even if Rorer's note was genuine he may simply have assumed that is what happened, since the Statutes of the University (1508) required that notice of academic disputations be given by fixing notices to the doors of all the churches in Wittenberg (although he could have asked Luther). Academic disputations, on a variety of subjects, were very much a part of university life, and proposing controversies for debate was not so much a privilege of a professor, as a duty. Whether he would personally attach them to doors or ask a university servant to do it is another matter.
Until the mid-20th century Melanchthon's version was simply accepted. In 1959 historian Hans Volz suggested that the theses may have been sent only to a select group on October 31st and published somewhat later. Then the Roman Catholic historian Irwin Iserloh, pointed out that there is no record Luther ever mentioned posting the theses on the church door, and proposed the idea that Luther never did publish his theses on the door, but that this was a later misunderstanding by Melanchthon. Coming shortly before the 450th anniversary, this proved controversial.
One issue here is whether Luther meant to start a formal local academic disputation, allowing others to write in if living at a distance, or whether he was, in fact, doing something different. Ordinarily an academic disputation should have been within a week and there is no record of any such disputation actually taking place. Luther, in a letter, said he had meant only to send them to a few for comment, in the context of explaining why he did not send a copy to someone in particular. This may indicate he had not intended, at that stage, to publish even in Wittenberg.
Luther's reservations about indulgences were already known in Wittenberg. He had previously preached on them. At least as regards Tetzel, the Wittenberg authorities was united in regarding him as a grave threat. The Elector Frederick had accumulated a very large collection of relics and these were exhibited in the Castle Church, for a fee, twice a year. This was on All Saints Day and Low Monday (the 9th day of Easter). Venerating such a large number of relics at once was held to bring huge benefit in remitting any temporal penalties of sin remaining after repentance, confession and absolution. There were so many relics that it is said venerating them together was equivalent to doing penance for 1,902,202 years and 270 days. Frederick had invested a great deal of money in accumulating the relics, with the intention of providing a source of income for the university he had founded in Wittenberg. Wittenberg did very well from the twice yearly pilgrims who came to see the relics, and the locals understood the religious imperative to repent, confess and receive absolution prior to viewing them.
The sale of indulgences was not allowed in the territory of the Elector Frederick. Nevertheless locals could nip across the border to buy them, returning to inform Luther and other local clergy they no longer needed to repent. Not only were Tetzel and his associates corrupting the locals, but pilgrims from elsewhere no longer needed to come to Wittenberg to see the relics. Why bother going to see the relics if one could buy full remission from Tetzel or his associates? Although Luther's reforms eventually led to the end of the exhibition of the relics in 1522, in 1517 it was Tetzel, not Luther, who was damaging the income stream from Wittenberg's exhibition of the relics. Anything against Tetzel, and the sale of indulgences, was exactly what the Wittenberg authorities wanted to hear. The date of October 31st was no random date.
Whether Luther intended to publish his theses in Wittenberg, within weeks they had "gone viral", and were printed and circulated all over Germany, and even beyond. Many were concerned about the amount of money leaving Germany for Rome (and elsewhere), and only too keen to see a religious objection to the practice. Luther said he was surprised by the reaction.
If Luther did personally attach the theses to the door it might not have seemed, in itself, of fundamental importance but merely fulfilling a university rule. The idea of the little monk bravely walking up to the door and posting his theses to the immediate condemnation of the entire church is a myth, but whether he did or did not fulfil the university regulation (if it applied to what he was about) is unknowable.
Leppin and Wengert conclude that:
When one sets aside emotion and the iconography connected with this debate, it becomes clear there are equally good arguments for and against the posting of the theses. It is a historically interesting but ultimately unresolvable question.
and
Whether or mot the theses were posted, the Reformation loses none of its historic importance.
The nailing of the theses continues to be a popular theme, as on this UK banner.

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