This question is expressly referring only to a narrow portion of the US agreements that are considered treaties under international law (per user102008's comment/answer etc.) But for simplicity henceforth I'll use "treaty" below to refer just these treaties ratified by a 2/3 majority of US senate, i.e. use US law terminology.
From a 2001 study by the Congressional Research Service... starting with a vague reference to Vattel (Law of Nations) and similar monist ideas:
As a general rule, international law and domestic law regarding
the amendment or modification, extension, suspension, and termination
of treaties and other international agreements are in substantial
It can be argued that amendment or modification, extension, suspension,
and termination of a treaty are essentially the forging of
new agreements and that, therefore, each is subject to the same
rules as apply to the making of a treaty, that is, conjoint action by
the President and the Senate. However, that conclusion is not established
by an unbroken line of consistent practice. By and large
the participation of the Senate with respect to amendment or modification
and extension of treaties seems fairly well established; suspension
seems largely left to Presidential determination; termination
has happened in such a variety of ways that it has been
said that ‘‘[n]o settled rule or procedure has been followed.’’
The last bit is footnoted to Whiteman, Marjorie. Digest of International Law, 1970. v. 14, 460 (hereafter cited as 14 Whiteman). Compare S. Rept. 97, 34th Cong., 1st Sess. See, generally, U.S. Congress. Senate. Committee on Foreign Relations. Termination of Treaties: The Constitutional Allocation of
Power. Committee Print. 95th Cong., 2d Sess. (1978).
And a bit later is detailed (not much) as:
The Constitution is silent on procedures for modifying or terminating treaties, and agreement has not been reached between the branches on a single proper mode. [...]
Twice in recent years the method of terminating a treaty has raised serious controversy within the United States. In 1978, President Carter terminated the defense treaty with the Republic of China [Taiwan] without the concurrence of either the Senate or Congress when he established diplomatic relations with the People’s Republic of China.
In 1977, the new Panama Canal Treaty terminated the 1903, 1936, and 1955 treaties with Panama. Although a new treaty was approved by the Senate, some contended that the termination of the earlier treaties required an act of Congress, thus including approval by the House of Representatives as well as the Senate.
Most of the discussion there is about modifications; I've quoted only the portion about exit/termination.
In practice, some (more recent) treaties have a clause that allows the US president to terminate them unilaterally, e.g. KORUS has/had:
If the President does invoke Article 24.5, and no further action is taken, KORUS will terminate 180 days after such notice is given.
Since the whole text is approved by the Senate, such an exit clause is basically like a mini "congressional-executive agreement" just on the exit route, in the sense that the Senate explicitly delegates the power to exit (a particular treaty) to the President.
And stated as a more general observation in the Congressional study
Termination.—At the international level, treaties often contain
provisions regarding duration and the method of termination,
or nations may terminate treaties by mutual consent. Grounds for
termination include violation of the agreement, but violation does
not automatically terminate a treaty.
Domestically, the Constitution does not prescribe the process for
the United States to terminate a treaty, and the process continues
to be controversial. Treaties have been terminated in a variety of
ways, including by the President following a joint resolution of
Congress, by the President following action by the Senate, by the
President and with subsequent congressional or Senate approval,
and by the President alone.
And the study gives examples of each later on...
executive withdrawal or termination pursuant to prior authorization or direction from Congress; [examples are fairly numerous, a recent one:] Comprehensive Anti-Apartheid Act of 1986, mandating that “[t]he Secretary of State shall terminate immediately” a tax treaty and protocol with South Africa
executive withdrawal or termination pursuant to prior authorization or direction from the Senate [most examples are old; the most recent I've seen mentioned: Wilson's termination in 1921 of the International Sanitary Convention of 1903]
executive withdrawal or termination without prior authorization, but with subsequent approval by Congress: [the only clear case given] "In 1864 the Secretary of State directed the U.S. Minister in London
to give the British Government the stipulated 6-months’ notice
of an intention to terminate the Great Lakes Agreement of 1817
regulating armaments on the Great Lakes. The minister did so,
and a few months later Congress by joint resolution ‘‘adopted and
ratified’’ the notice of termination [There's a confusing case in 1911, in which Congress "adopted and ratified" a declaration of President Taft terminating a 1832 treaty with Russia, which the President only asked the Senate to ratify].
executive withdrawal or termination without prior authorization, but with subsequent approval by the Senate; [no real examples except the confusing one with Taft (from right above), interpreted differently]
unilateral executive withdrawal or termination without authorization or direction by Congress or the Senate [famous example(s):] Telegram from the U.S. Department of State to the Embassy of the Republic of China (Dec. 23, 1978) [leading to Goldwater v. Carter, in which the Supreme Court declined to rule on the constituionality of the Telegram but also dismissed the complaint against it. Based on this precedent, Reagan unilaterally terminated a Treaty of Friendship, Commerce and Navigation with Nicaragua, and the (district) courts also dismissed the complaint against this termination, citing Goldwater v. Carter. Then came Bush and the ABM Treaty, the complaint against it dismissed in Kucinich v. Bush.]
Who in the US determines treaty violation(s) so bad that the US can exit the treaty is basically the same issue as who can decide termination (which is why I mentioned that first.) And since that's the case, what can also happen in case of violations is...
Suspension.—The President conveys notice of suspension of a
treaty and makes the determination that would justify suspension,
such as a fundamental change in circumstances or material breach
of a treaty by another party.
(Giving examples of suspension is veering too much off-topic.)
And regarding ultra vires (void treaties): following Missouri v. Holland (1920) and Reid v. Covert (1957), the US Constitution (unsurprisingly) was found to take precedence and void any treaty (or agreement) clauses violating it; quoting from the latter case:
[n]o agreement with a foreign nation can confer power on the Congress,
or on any other branch of Government, which is free from the restraints of the Constitution.
In Missouri v. Holland, the Supreme Court found nothing violating the Constitution in the treaty in question (on migratory birds). Reid v. Covert was about an executive agreement (not treaty) that was found unconstitutional because it allowed US citizens who were not members of the military to be judged abroad by military tribunals. I'm not aware of cases after that... but there were before Reid v. Convert basiscally overturned In re Ross (a case from 1891) as "a relic from a different era"; in that case a seaman was tried abroad by a consular court pursuant to a treaty.
Since the 1960s, virtually all examples of such tension between treaty and Constition have been issues handled at accession rather than being discovered thereafter:
Treaty practice continues to be consistent with Reid v. Covert. Where accession to an international treaty regime poses a plain downward departure from internal interpretations of individual rights, the United States has rejected or qualified its participation.
One more thing to note is that the US has singed (in 1970) but not ratified the Vienna Convention on the Law of Treaties. The convention is still on the pending treaties list of the Senate and successive US administrations have stood by the signature when queried explicitly.
The Convention is somewhat relevant here because it also covers treaty termination routes,
[a]ny act declaring invalid, terminating, withdrawing from or suspending the operation of a treaty pursuant to the provisions of the treaty ... through an instrument communicated to the other parties. If the instrument is not signed by the Head of State, Head of Government or Minister for Foreign Affairs, the representative of the State communicating it may be called upon to produce full powers.
But this text largely ignores everything particular to the US, e.g. one commentary is:
Under this rule, a notice of withdrawal issued by the President (i.e., the “Head of State” for the United States) would effectively withdraw the United States from the international agreement as a matter of international law, providing such notice complied with applicable treaty withdrawal provisions. In this regard, the withdrawal process under international law may not account for the unique constitutional and separation of powers principles related to withdrawal under U.S. domestic law.
The convention also covers disputes, but those are mostly referred to the ICJ and since 1985 the US has decided not to recognize the ICJ's authority except on a case-by-case basis.