Step 1: Identify the noun used in the sentence.
“Vaigai Express” is the name of a specific train.
Step 2: Rule of articles for proper and specific nouns.
The definite article “the” is used:
Before names of trains, ships, and newspapers,
When referring to a particular, well-known object.
Step 3: Apply the rule to the sentence.
Since “Vaigai Express” refers to a particular train, the correct sentence is:
\[
\text{I go to Madurai by the Vaigai Express.}
\]
Step 4: Eliminate other options.
a, an: used for non-specific singular nouns
some: used with plural or uncountable nouns
Hence, the correct answer is \(\boxed{\text{the}}\).