Under which Seal does the Rapture of the Church take place?
According to this teaching, the Rapture of the Church takes place under the Sixth Seal, which serves as a preview of the Trumpet and Vial Judgments, including the sealing of the 144,000 and the rapture of the Church.
The Sixth Seal is different from all the others. While the first five seals introduce individual entities and events, the Sixth Seal gives a summary overview of everything contained in the rest of the book of Revelation, including the Trumpet and Vial Judgments and the events leading into the Millennial Reign of Christ. It is within this seal that the great earthquake happens, the sun turns black, the moon becomes as blood, the stars fall, and the heavens depart as a scroll. The people of the earth then cry out that the Day of the Lord's wrath has come, though the wrath itself has not yet been poured out.
Before that wrath can actually fall, two events must still take place, and both are shown within the scope of the Sixth Seal: the sealing of the 144,000 servants of God, and the rapture of the Church, since the Church is not appointed to the wrath of God. Revelation 7 shows the angel commanding that the earth not be hurt until the 144,000 are sealed, and immediately following that, the great multitude which no man can number stands before the throne, clothed in white robes, crying, Salvation to our God which sitteth upon the throne, and unto the Lamb. This great multitude is identified as those who came out of great tribulation, having washed their robes in the blood of the Lamb, which is understood as the raptured Church.
The teaching lays this out plainly in its summary of the Sixth Seal: Great Earthquake, Sun Darkens, Moon Turns to Blood, Stars Fall to Earth, Rapture of 144,000 Jewish Virgins, and Rapture of the Church. So while the seals themselves are not the actual events but rather serve like a table of contents announcing what is to come, it is specifically under the Sixth Seal that the Rapture of the Church is announced and previewed, in connection with Matthew 24:29-31 where the sun is darkened, the stars fall, and the Son of man sends His angels with a great sound of a trumpet to gather together His elect.
The scriptures
And I beheld when he had opened the sixth seal, and, lo, there was a great earthquake; and the sun became black as sackcloth of hair, and the moon became as blood; And the stars of heaven fell unto the earth, even as a fig tree casteth her untimely figs, when she is shaken of a mighty wind. And the heaven departed as a scroll when it is rolled together; and every mountain and island were moved out of their places. And the kings of the earth, and the great men, and the rich men, and the chief captains, and the mighty men, and every bondman, and every free man, hid themselves in the dens and in the rocks of the mountains; And said to the mountains and rocks, Fall on us, and hide us from the face of him that sitteth on the throne, and from the wrath of the Lamb: For the great day of his wrath is come; and who shall be able to stand?
And after these things I saw four angels standing on the four corners of the earth, holding the four winds of the earth, that the wind should not blow on the earth, nor on the sea, nor on any tree. And I saw another angel ascending from the east, having the seal of the living God: and he cried with a loud voice to the four angels, to whom it was given to hurt the earth and the sea, Saying, Hurt not the earth, neither the sea, nor the trees, till we have sealed the servants of our God in their foreheads. And I heard the number of them which were sealed: and there were sealed an hundred and forty and four thousand of all the tribes of the children of Israel. Of the tribe of Juda were sealed twelve thousand. Of the tribe of Reuben were sealed twelve thousand. Of the tribe of Gad were sealed twelve thousand. Of the tribe of Aser were sealed twelve thousand. Of the tribe of Nephthalim were sealed twelve thousand. Of the tribe of Manasses were sealed twelve thousand. Of the tribe of Simeon were sealed twelve thousand. Of the tribe of Levi were sealed twelve thousand. Of the tribe of Issachar were sealed twelve thousand. Of the tribe of Zabulon were sealed twelve thousand. Of the tribe of Joseph were sealed twelve thousand. Of the tribe of Benjamin were sealed twelve thousand. After this I beheld, and, lo, a great multitude, which no man could number, of all nations, and kindreds, and people, and tongues, stood before the throne, and before the Lamb, clothed with white robes, and palms in their hands; And cried with a loud voice, saying, Salvation to our God which sitteth upon the throne, and unto the Lamb. And all the angels stood round about the throne, and about the elders and the four beasts, and fell before the throne on their faces, and worshipped God, Saying, Amen: Blessing, and glory, and wisdom, and thanksgiving, and honour, and power, and might, be unto our God for ever and ever. Amen. And one of the elders answered, saying unto me, What are these which are arrayed in white robes? and whence came they? And I said unto him, Sir, thou knowest. And he said to me, These are they which came out of great tribulation, and have washed their robes, and made them white in the blood of the Lamb. Therefore are they before the throne of God, and serve him day and night in his temple: and he that sitteth on the throne shall dwell among them. They shall hunger no more, neither thirst any more; neither shall the sun light on them, nor any heat. For the Lamb which is in the midst of the throne shall feed them, and shall lead them unto living fountains of waters: and God shall wipe away all tears from their eyes.
Immediately after the tribulation of those days shall the sun be darkened, and the moon shall not give her light, and the stars shall fall from heaven, and the powers of the heavens shall be shaken: And then shall appear the sign of the Son of man in heaven: and then shall all the tribes of the earth mourn, and they shall see the Son of man coming in the clouds of heaven with power and great glory. And he shall send his angels with a great sound of a trumpet, and they shall gather together his elect from the four winds, from one end of heaven to the other. Now learn a parable of the fig tree; When his branch is yet tender, and putteth forth leaves, ye know that summer is nigh: So likewise ye, when ye shall see all these things, know that it is near, even at the doors.
For God hath not appointed us to wrath, but to obtain salvation by our Lord Jesus Christ,
From the books
This answer is drawn from these passages in the Mary Banks Faith Library.