You do know how to ask difficult questions don't you?
First thing to say is synthetics vary a great deal - some are more stable than most essential oils, some much less so. For example some of the synthetic musks are so stable that they have given rise to environmental concerns about their longevity in the environment. On the other hand some of the aldehydes are the things most often responsible for a modern perfume 'going off' - that vinegary smell you sometimes get in a bad perfume is usually from the aldehydes in it that have oxidised. Perfumes that have a lot of bright, sparkling, fruity notes in them such as Channel No 5 and many fragrances aimed at the Brazilian market are particularly high in these aldehydes.
They can certainly boost essential oils in lots of ways - extending the citrus scent is a classic role for synthetics in a blend for example - some are used to 'exalt' the florals: materials like Hedione have little perfume on their own but have a profound effect on other components, especially florals, in a blend. Still others like Iso E Super have a filling, rounding effect and velvet feel that is hard to replicate with anything natural but which works together with natural components to give something exceptional.
The classic blend for a Fougère style is oakmoss, lavender, coumarin and (usually) geranium: the original of the style Fougère Royal was the first fragrance known to use a synthetic component (the coumarin) and would not be the same without it. Equally it wouldn't be the same without the oakmoss but now that oakmoss is so heavily restricted by IFRA it is becoming necessary to replace it with synthetic substitutes such as Veramoss.
And that brings me to the biggest and most controversial use of synthetics, to replace naturals. Sometimes this is just down to price and even the small-scale DIY enthusiast may be tempted to go that route faced with the enormous prices of some absolutes or with oils such as sandalwood.
Another reason for replacement is the environmental and moral problem that comes with using the naturals: Santalum album is an endangered species: many would argue it is irresponsible to use what remains to make oil for perfume - there are natural and synthetic alternatives and both have their merits and problems.
Replacing animal materials is another major area for synthetics: all the traditional animal ingredients have synthetic alternatives and price, morality and public anxiety have all played a part in driving the industry down the synthetic route.
I think I've put a list up on this forum before of synthetics I consider good ones to get to know first - that list is probably similar to the answer to your final question, though of course as formulae are generally closely guarded, getting to know who uses what isn't straightforward. I have nearly 500 ingredients in my collection now, just over half of which are synthetic: there are thousands more that I don't keep stock of and new materials are becoming available all the time. I'll track down the list and put it into this thread too.