In Colossians 2:18, Paul warns his readers, "Do not let anyone who delights in false humility and the worship of angels disqualify you for the prize" (NIV emphasis mine). Of course, that it is in the genitive case means that we can take in two ways. Either it refers to worship directed to angels or worship done by angels. Is there any reason to prefer one or the other here?
Hard Question, Soldarnal
Peter O'Brien says of this verse in his Colossians commentary:
This verse has been described as one of the most contested passages in the NT, presenting great difficulties in language and content.
And Douglas Moo, in his:
This verse furnishes the most important evidence about the false teaching, but it is also arguably the most difficult verse in Colossians to interpret.
(The difficulties of this verse of course go far beyond this particular phrase.)
Both interpretations have been argued. If the subjective genitive position (angels worshiping God) is taken, the idea would be that the angels are mediating in some sense for the humans.
The objective genitive is the more obvious and traditional reading of the text.
The phrase has normally been taken to denote "the worship directed to the angels." —O'Brien
However, O'Brien concludes after a complex discussion that
the false teachers claimed to have joined in the angelic worship of God as they entered into the heavenly realm and prepared to receive visions of divine mysteries.
Which I frankly think is convoluted. (O'Brien is thorough but sometimes prefers esoteric renderings.) Good ol' Moo comes to the rescue, giving a simple rebuttal of O'Brien's complex discussion. It is worth quoting at length:
The more traditional interpretation should probably be preferred. First, from a purely linguistic point of view, the phrase is more likely to mean "worship offered to angels" than "worship offered by angels." Second, while many Jewish apocalyptic texts refer to angelic worship, and some refer to humans observing or imitating that worship, very few speak of humans joining with angels in such worship. It is questionable, in other words, whether the simple (and very ambiguous) phrase "worship of angels" would have been capable of connoting such a relatively rare concept. Third, as we have noted, a key concern of Colossians has been to accentuate the superiority of Christ over spiritual beings (1:6, 20; 2:10, 15). Such a concern to minimize the significance of the angels would make very good sense if, indeed, the false teachers were worshiping them. Fourth, Clinton Arnold has suggested a plausible background for Paul's accusation that the false teachers were worshiping angels. He notes the importance of invoking angles as a means to ward off evil in the ancient world in general and the geographic region of Colossae in particular. Paul would be characterizing this calling on angels for protection as a tantamount to the worship of angels.
As an aside, Richard Gamble makes an argument from Hebrews 9-10 that Christians do actually join with angels in the worship of God, which makes the subjective genitive seem even less likely.
"Worship of angels" indicates men worshiping angels.
KaZark answered this well.
I am just throwing some common sense to support his answer. If you notice how people like to worship Saints in our day , it's not hard to imagine before Christ people liked to worship angels. When you consider how the Law was delivered by angels (Heb 2:2) and that in the Old Testament angels appearing was the greatest thing ever. Naturally the fleshly mind would have invented a host of superstitions about different angels, their names, how you could invoke their blessing, etc. they were probably all puffed up making pretend rules about things they no nothing of. They probably had various fake practices of humility in the way they pretended to be religious in their superstitions. Apparently even many Jews must have had a very high view of angels and commonly fell into this kind of thinking, for Hebrews starts by arguing how Jesus is superior to angels. If we could argue to superstitious people of our day who prefer to go to Mary, or St. so-and-so, rather than the eternal Priest, who knows how to empathize with our weakness, we would probably find very similar attitudes and ideas. Then it was angels, now it is Saints.
Let no one defraud you of your price – the price of everlasting life in heaven.
Colossians 2:18-19 (NASB)
18 "Let no one keep [a]defrauding you of your prize by delighting in [b]self-abasement and the worship of the angels, [c]taking his stand on visions he has seen, [d]inflated without cause by his fleshly mind,"
19 "and not holding fast to the head, from whom the entire body, being supplied and held together by the joints and [e]ligaments, grows with a growth [f]which is from God."**
This is a contextual interpretation of the question (What is the “worship of angels?), from the passage Colossians 2:18 -23.
Paul is warning the Colossians not to be led astray by the deceptive arguments in support of the “worship of angels” that the false teachers are trying to impose on the Christian church of Colossae and surrounding churches, and gives a character profile of the false teachers.
1 / “self-abasement/mock worship and the worship of the angels,”
The false teacher takes pleasure in showing off fasting and other denials,
Matthew 6:16 (NRSV
“And whenever you fast, do not look dismal, like the hypocrites, for they disfigure their faces so as to show others that they are fasting. Truly, I tell you, they have received their reward."
Worship of angels.
It is apparent that Christians converted from Judaism, believed in “angel worship”, saying , after all did, not Daniel saw thousands of angels ministering God and 100 million standing in front of him (Daniel 7:10) and did not Jacob wrestled with an angel all night ? (Gen.32:24-32) Mike above also mentioned that angels delivered the law. (Heb.2:2)
Obviously, these Judaic Christians wanted to infuse “angel worship” into Christianity .How?
2 / By “Taking his stand on visions he has seen,” ,
Angel worship is forbidden in the scriptures, (Compare 2 Tim.2:5; and Rev. 19:10)regardless of this the false teachers ignore the scriptures by, “taking his stand on visions he has seen,” claiming that he has supernatural insights and visions that surpass the teachings of the scriptures. Moreover, telling Christians that it is not possible to worship God directly, you must be humble and worship Him thru lower beings.
Similarly today, some believe in “saints’ worship”. God is an unknown quantity to many Christians and believe He cannot be reached directly, so they worship him, thru intermediate beings, saying are closer to God in heaven, saints today, and angels in Paul’s days.
3/being “inflated without cause by his fleshly mind,”
4/ “and not holding fast to the head, Jesus Christ.”
Angel worship is flatly forbidden in the scriptures 2Tim.2:5 and Rev.19:10
Taking the phrase “worship of angels” out of context, it may be interpreted in two ways, “worshipping angels” or worship instituted by angels. By doing this the false teachers , ignore Pauls’ powerful letter to the Colossians to be aware of the false teachers, with their puffed up piety caused by his fleshy mind and outward superior reverence of God.
Wikipedia has this under "Demiurge":
Angels Psalm 82 begins (verse 1), "God stands in the assembly of El (the Septuagint here says the assembly of gods), in the midst of the gods he renders judgment", indicating a plurality of gods, although it does not indicate that these gods were co-actors in creation. Philo had inferred from the expression "Let us make man" of the Book of Genesis that God had used other beings as assistants in the creation of man, and he explains in this way why man is capable of vice as well as virtue, ascribing the origin of the latter to God, of the former to His helpers in the work of creation.
The earliest Gnostic sects ascribe the work of creation to angels, some of them using the same passage in Genesis. So Irenaeus tells of the system of Simon Magus, of the system of Menander, of the system of Saturninus, in which the number of these angels is reckoned as seven, and of the system of Carpocrates. In the report of the system of Basilides, we are told that our world was made by the angels who occupy the lowest heaven; but special mention is made of their chief, who is said to have been the God of the Jews, to have led that people out of the land of Egypt, and to have given them their law. The prophecies are ascribed not to the chief but to the other world-making angels.
The Latin translation, confirmed by Hippolytus of Rome, makes Irenaeus state that according to Cerinthus (who shows Ebionite influence), creation was made by a power quite separate from the Supreme God and ignorant of Him. Theodoret, who here copies Irenaeus, turns this into the plural number "powers," and so Epiphanius of Salamis represents Cerinthus as agreeing with Carpocrates in the doctrine that the world was made by angels.
It seems to me that with the angels being held as creators and co-creators it is a small step to their offering them obeisance of some kind. I do like to note that in verse 20 that they understood the Angel of the LORD to be a single, chief angel who led the Jews out of Egypt:
 we are told
that our world was made by the angels who occupy the lowest heaven; but special mention is made of their chief, who is said to have been the God of the Jews, to have led that people out of the land of Egypt, and to have given them their law.** The prophecies are ascribed not to the chief but to the other world-making angels.
One must note that mainstream Christian groups seem to also understand the Angel of the LORD to be the "pre-incarnate Christ" though they don't seem to be consistent.
St.Paul warns about worshiping God's servants, because God's servants are equal to angels. Ref: Zechariah 12:8. St.Paul clearly not mentioning about angels of heaven, for example Gabriel, Michael and others and as well as not about the fallen angels too. Because it was addressed to Colossians who are strong in doctrine. Today too believers are fond of worshiping such angels as the worldly people worships Singers, Sportsmen and Movie stars. Instead of honoring and listen to their doctrine these people start to worship them and unfortunately some of the God's servants too allow them unknowingly or wantonly. Such acts bring God's wrath on them.