Hi James, you can also turn the sentence round:

El coche se me ha averiado.

In any case, if in the other sentence you include "me" it sounds better the other way round:

No me arranca el coche.

so with "mi coche no arranca" you're using a possesive pronoun and when you say it the other way "No me arranca el coche" the "me " is an indirect or direct pronoun ? These pronouns give me some trouble.

I think you can use the me instead of the possessive pronoun in both cases.

Mi coche no arranca.
El coche no me arranca.
No arranca mi coche.
No me arranca el coche.

As to the word order, the part that comes last is generally given more emphasis in Spanish.

James and James: the possessive adjective.

Me =here, indirect object.

You have possesive adjectives and pronouns, so they both are generally referred to just as "possisives".

Spanish is flexible when it comes to word placing in a sentence. However, when you use indirect object pronouns to indicate that something is affecting you, it will appear before the verb, so in these cases it is somewhat more common place the subject at the end, leaving the real concerned human at the beginning of the sentence. Unless you want to stress the importance of the object, and leave yourself on a secundary plane of interest in the sentence, of course.

Me gusta algo.
Algo me gusta.

Se me ha estropeado el coche.
El coche se me ha estropeado.

El coche no (me) arranca.
No (me) arranca el coche.

Of course, they are all correct.