If you say, "The Great Storm of 1987", that implies that there is one particular storm that happened that year that was particularly notable. I wouldn't necessarily expect everyone in the world to know what storm you're talking about, but almost everyone who lives in the area affected by it would know. At that point "the Great Storm of 1987" is like a proper ...
It refers to a particular great storm that the speaker expects his listener to know about. If it were just any storm, then "a big storm in 1987" would fit. The word "great" isn't usually used to refer to just any storm. (Consider, for example, the "Great Flood of 1993" in the midwestern U.S.)
Now, there are more than a few events that merit the term "great", ...
This is an error. The error seems to be quite common, perhaps as people write "an important" then edit to change "important" to "significant", but forget to change the article. But this is speculation. Nevertheless, it is an error. You shouldn't say (or write) "an significant".
The word "an" is based entirely on the vowel sound of the following word, ...
The labeling on the Cambridge dictionary website is slightly confusing. Looking at the definition for worsening, we see it is labeled "U or S".
"U" means "Uncountable or singular noun: a noun that has no plural." I think it's saying that if a noun is uncountable, it must be singular. This is true. There is no such thing as a plural uncountable.
I don't know that there is a universal rule here, however, I would lean towards using "the" as man-made structures frequently use it.
The Empire State Building
The Taj Mahal
However, there are plenty of examples that don't use it.
1. I went to Wrigley Field to watch the Cubs play
2. I go to church at Saint John The Baptist Catholic Church
In this case, the article is optional.
The response is idiomatic both with and without the a. The choice might depend on exactly what's in the respondent's mind when s/he answers.
If the respondent imagines choosing a number from the numbers presented, the answer is likely to be a nine. Similarly a contestant in a TV quiz might choose the (number) nine ...