What is the corruption Mormons believe will infect their church before the second coming?

As far as I know, this passage of scripture has not been widely emphasized or expounded by the prophets, although there are hints here and there. Various apostles have recently referred in general terms of hardships yet to to come.

In the Book of Mormon, (3 Ne 16) the resurrected Christ gives a warning to the unbelieving Gentiles that they would be destroyed should they reject the Gospel, while those who accepted it would be counted among righteous Israel. 16:10 seems to address the possibility that some would accept it and then turn away. The Savior returned to this theme of destruction of Gentile wickedness when he came a second day to a larger congregation and expanded on it, (3 Ne 20-21), connecting it to prophecies of Isaiah, and also suggesting that converted Lamanites would play a primary role on building the New Jerusalem, with faithful Gentiles assisting.

The Doctrine and Covenants mentions (1:16) that those who "will not hear the voice of the Lord, neither the voice of his servants, neither give heed to the words of the prophets and apostles, shall be cut off from among the people". No differentiation between member and nonmember is made and no time frame is specified.

There is a prophecy attributed to Heber C. Kimball (A contemporary of Brigham Young) describing a future period of persecution and at least partial apostasy. Recognizable fragments have been widely and informally circulated through LDS culture since his son J. Golden Kimball presented it (Conference Report, Oct. 1930, p. 58-59) although I am unaware of earlier text or later exposition.

Although the leaders of the Church have repeatedly affirmed that there will not be another great apostasy resulting in the loss of the entire church, they have never specified how many will remain faithful when the rest of the world is descending into greater wickedness. It is possible to imagine various reasons they might be circumspect about reference to prophecies of apostasy (even incomplete or partial) and consequent destruction, but the scriptures are there to be read.

There are no few Latter-day Saints who see the various criticisms of the Church and its leaders by a few dissidents, including members and former members, over matters such as feminism and gay rights, as just the kind of corruption the OP refers to.

This part of this revelation speaks about 2 things: Darkness and corruption and Vengeance. This part of section 112 is not taking about corruption in the church today, it's talking about the state of the world when the church was organized. It is not a prophecy about the spreading of corruption in the church, rather the coming of judgement to the world.

Darkness and Corruption

Regarding the darkness and corruption, Joseph Fielding Smith clarified this as referring to the spiritual darkness and corruption that existed at the time of the revelation:

"This proclamation should cause us even in this day serious reflection. If darkness covered the earth in 1837, surely it has deepened in its blackness since that day. If that was a day of wickedness, and the Lord in several revelations testified to this fact, then it is even more so today." [Church History and Modern Revelation, 2:74]

We know that after the death of Christ, the apostles and the church were persecuted and killed to the point where there would be a "famine of the word," Amos 8:11, see Preach My Gospel. This loss of spiritual light, and the revelation from the living prophets left a void of which God declared,

"all their creeds were an abomination in his sight; that those professors were all corrupt; that: “they draw near to me with their lips, but their hearts are far from me" JSH 1:19

Vengeance

Without the fullness of the law, however, the Lord does not hold men accountable,

"He will judge them, not according to what they have not, but according to what they have" Teachings of the Presidents of the Church: Joseph Smith Ch. 35

So when the Lord speaks of vengeance going forth from his house, it refers to the restoration of the gospel in this dispensation:

"...it is a dispensation in which the Gospel has been revealed from heaven, the servants of God called to labor in the vineyard for the last time, and in which the Lord intends to pour out great and terrible judgments upon the nations of the wicked after they have been warned by the sound of the everlasting Gospel." - [Latter-Day Judgments. Discourse by Elder Orson Pratt, delivered in the 13th Ward Assembly Rooms, Salt Lake City, Sunday Afternoon, January 26, 1873.]

Naturally, the first to receive these judgements would begin with the first to reject the restored gospel:

25 And upon my house shall it begin, and from my house shall it go forth, saith the Lord;

26 First among those among you, saith the Lord, who have professed to know my name and have not known me, and have blasphemed against me in the midst of my house, saith the Lord.

This behavior matches the definition of personal apostasy: "turning away from the Church and ultimately denying the faith.." Teachings of President Brigham Young

What do the scriptures and/or Church leaders suggest is the corruption the Lord is speaking about?

Elder Claudio D. Zivic in a talk titled Avoiding Personal Apostasy, mentioned a few things that this could include (though surely not exhaustive):

• When a person commits serious sin but does not repent
• Conflicts between Church members, which includes being offended by leaders and other members.
• Faultfinding, including beginning to think one could make better decisions than our leaders
• Assuming authority one does not possess or when one seeks revelation for a stewardship outside of their sphere of responsibility