By a Sign I mean when some happening is intended - by God - to teach us something.
Since this is the function of a Sign, typically the fact of it - its 'form' - is the key to its importance; while the specific 'content' of the Sign is much less important.
Synchronicity describes a person's experience of coincidences that seem so unlikely as to seem meaningful. Calling it Synchronicity means that we do not believe it is a coincidence. And if not a coincidence, then it is planned.
Synchronicity could be regarded as a sign that there is a creator, and this creator is God.
Why? Because the organization of many events to create such coincidences implies that there is a purposive creator, with a unified will; and the fact that the coincidences are personally directed - that we notice them - suggests that the creator is himself a person (i.e. God); and concerned with individual persons.
(Because why would an impersonal 'deity' arrange Synchronicities?)
In other words, the fact that I experience synchronicities (if I accept it as a fact), implies a planner - and what is more a planner who takes a me-focused view of the world.
But a Synchronicity may be - often/ usually is - something apparently meaningless; or at least trivial. Therefore the significance is in the fact of Synchronicity - not the content of it. Thus Synchronicity is a Sign - designed to teach us some particular thing - but not to teach us everything!
Serendipity could be regarded as a sub-type of Synchronicity when the coincidences make a 'happy accident'.
Examples of Serendipity include when a scientist makes a fortunate discovery based on some unplanned event - such as Alexander Fleming's happy accident in the discovery of penicillin.
But fortunate coincidences happen in everyday life - when someone arrives 'just in time' without knowing in advance that time was critical; when someone was in the right place at the right time; or happens to be carrying just exactly what would be needed... etc.
Serendipity can be considered a sub-type of Synchronicity - a Synchronicity that has fortunate consequences.
Serendipity could therefore be regarded as a Sign that, not only is there a creator God; but that this God loves us personally.
Serendipity may therefore be understood as a Sign of the working of the Christian God - or a God much like Him. Not only does He personally arrange events around us to make remarkable coincidences; but these are fortunate coincidences.
In other words; 'happy accidents' are signs of the truth of the Christian God.
But Synchronicity and Serendipity only work as Signs for those who regard Signs from God as a possibility.
An atheist/ materialist who assumes that there is only impersonal causation and random chance, can never be convinced of a Serendipity - he has ruled-out the possibility, by his assumptions.
Signs therefore do not prove anything - because what counts as proof requires metaphysical assumptions concerning what is real.
So, if we have already decided (perhaps implicitly, without being aware of this prior decision) against God - that there is no God, or cannot-be God, or that a deity cannot be personal. or if we had re-decided that it is silly wishful thinking to suppose that God might arrange a happy 'accident', a fortunate 'coincidence'...
Then when the possibilities of a loving, personal, creator God have already-been ruled-out; then clearly any Sign will be ignored, reinterpreted, explained-away.
And the intended teaching behind that Sign will not be regarded as purposive; so the chance to learn will be missed. Indeed - that there was a chance to learn will be missed.
Such is normal, mainstream and officially-approved behaviour - here-and-now.
Therefore, Synchronicity and Serendipity are Signs of the reality and nature of God; intended for those who are open to such possibilities.
Each instance is tailored for the needs and situation of each particular person. A Sign is 'for' a specific individual and circumstance - not intended as a general proof for everybody.
The lesson of Signs is to recognize them for what they are; to learn from what is intended by their form - and therefore not to get bogged-down on trying to interpret the specific details of the content.
Synchronicity is telling us, personally - here and now, that God is active in the world; it does not mean that what happens by remote coincidence is also a form of covert but specific life guidance. Even a serendipitous discovery in science may turn out later to be inaccurate, false - or some better theory may emerge later. The value of a happy accident is intended specifically for the time it happened and that person - not 'happiness forever', nor happiness for everybody.
If we begin to think in terms of Signs; if we focus on the forms rather than specific contents; we may find great encouragement from the daily events of our lives.
After all, personal re-affirmations of the reality of God, creator of this world, of His Goodness and Love, are of inestimable value.
One cannot get too-much of such things!
Note: I got this concept of a Sign from By the Hand of Mormon by Terryl Givens (2003); which is an historical examination of the way that the Book of Mormon functioned as a Sign for most of the history of the CJCLDS. Only recently (the past several decades) did the specific content of the Book - its scriptural teachings - become a focus. Up to the 1970s; the BoM functioned mostly as an instrument of conversion, as evidence of a new 'dispensation'; as evidence of God's active intervention in the modern world by personal revelations. It was more the circumstances and fact of the BoM that was important, and little attention was paid to the teachings that could be gleaned from its text. (Currently, however, Mormons tend to regard the BoM as a scripture on the same level, and to be studied in the same way, as The Bible.) This history clarified for me the nature of a Sign, and the kinds of things it could accomplish.