Validity of the August Weismann mouse-detailing experiment refuting Lamarckism
If you are reading this after 31st July 2017 it will seem somewhat disconnected from the question. This is because the question has been extensively edited since I wrote this answer.

Anyone who is interested in the work of August Weismann should start by reading the corresponding Wikipedia page.

Ernst Mayr ranked him as the second most notable evolutionary theorist of the 19th century, after Charles Darwin.

As far as the mouse tail experiment is concerned, we learn that Weismann was concerned with disproving Lamarckianism, and that:

Weismann was aware of the limitations of this experiment, and made it clear that he embarked on the experiment precisely because, at the time, there were many claims of animals inheriting mutilations (he refers to a claim regarding a cat that had lost its tail having numerous tail-less offspring).

My grandfather used to claim that his mother was frightened by a bear whilst pregnant, which was was why he was born with bare feet but that's more likely to be due to an epigenetic effect I suppose.

Added later in response to Remi.b
I see that Charles has already addressed this to some extent, but since I have already done the research I'll post what I found:
from August Weismann, Essays Upon Heredity. Clarendon Press, Oxford, 1889
(Volume 1, Chapter 8, p. 431)

But now, after this little digression, let us return to the transmission of mutilations.
We have seen that the rudimentary tails of cats and dogs, as far as they can be submitted to scientific investigation, do not depend upon the transmission of artificial mutilation, but upon the spontaneous appearance of degeneration in the vertebral column of the tail. The opinion may, however, be still held that the customary artificial mutilation of the tail, in many races of dogs and cats, has at least produced a number of rudimentary tails, although, perhaps, not all of them. It might be maintained that the fact of the spontaneous appearance of rudimentary tails does not disprove the supposition that the character may also depend upon the transmission of artificial mutilation.
Obviously, such a question can only be decided by experiment: not, of course, experiments upon dogs and cats, as Bonnet rightly remarks, but experiments upon animals the tails of which are not already in a process of reduction. Bonnet proposes that the question should be investigated in white rats or mice, in which the length of the tail is very uniform, and the occurrence of rudimentary tails is unknown.
Before this suggestion was made, I had already attacked the
problem experimentally. Such a course might, perhaps, have been more natural to those who maintain the transmission of mutilations, to which I am opposed. Although I undertook the experiments expecting to obtain purely negative results, I thought that the latter would not be entirely valueless; and since the numerous supporters of the transmission of acquired characters do not seem to be willing to test their opinion experimentally, I have undertaken the not very large amount of trouble which is necessary in order to conduct such an experimental test.