The status of a fetus below 40 days is seen in many cases as insignificant. The following are three major areas in Halacha where this insignificance can be found:
1) The laws of Terumah
A Bas Kohen who marries a Yisroel loses her right to partake of Terumah if she becomes pregnant by her husband. Suppose her husband dies on the day of their marriage. In that case, the din is that she may partake of Terumah up until 40 days because, as the Gemara learns, even if it turns out that she is pregnant from her late husband after 40 days, the status of the fetus within those 40 days is considered like mere water and would have no legal significance to her:
יבמות סט ע"ב
אָמַר רַב חִסְדָּא: טוֹבֶלֶת וְאוֹכֶלֶת עַד אַרְבָּעִים, דְּאִי לָא מִיעַבְּרָא — הָא לָא מִיעַבְּרָא, וְאִי מִיעַבְּרָא — עַד אַרְבָּעִים מַיָּא בְּעָלְמָא הִיא.
2) The laws of Tumah and Tahara
The Mishana in Maseches Nida rules that a woman who miscarried within 40 days need not worry about impurity caused by the miscarriage as the formation of the offspring in the womb occurs only forty days after conception:
נדה ל ע"א
מתני׳ המפלת ליום מ' אינה חוששת לולד ליום מ"א תשב לזכר ולנקבה ולנדה
3) The laws of Bechoros
The Mishna in Maseches Bechoros establishes that only a miscarriage over 40 days has the legal status of a firtsborn:
בכורות מז ע"ב
מתני׳ אי זהו בכור לנחלה ולכהן… והמפלת ליום ארבעים הבא אחריהם בכור לנחלה ולכהן
The Halacha varies in context of levels of danger to the mother and stage of the pregnancy.
Therefore, it is imperative to consult a Rov in all cases, whether is seems to be a case where one should be maikel or machmir.