Dropping the subject from sentences

Consider this example:

He got into the car. Started the engine, checked the mirrors. Stepped on the gas and headed down Main Street.

Omitting the subject from a sentence isn't proper construction, strictly speaking, but it infuses the writing with a sense of breathless urgency. I've seen a few writers use this style (notably in fast-paced suspense novels).

• How common is this style?

• Is there a name for it? (I was thinking staccato style, but that refers to short, choppy sentences in general)

Solution 1:

There is grammatical ellipsis, which in general case might not introduce the sense of accelerated time, but quite the reverse, depending on how hard is it to parse, for example:

The average person thinks he isn't.

The parsing here needs backtracking which slows the reading, but gives an effect of delayed and double, in-depth comprehension.

In terms of pure effect, look at diazeugma

The figure by which a single subject governs several verbs or verbal constructions (usually arranged in parallel fashion and expressing a similar idea):

The Romans destroyed Numantia, razed Carthage, obliterated Corinth, overthrew Fregellae.

Wikipedia defines Diazaugma Disjunction:

The subject appears at the beginning of the sentence and each verb follows in its respective clause.

EDIT:
As I said - for pure effect this is diazeugma, but the problem is that the text is agrammatical and the figures apply to grammatically correct texts. However, I think that is a narrow view. I find this text explains the matter in appropriate detail, from which I will only very shortly cite:

Almost 30 years ago, in An Alternate Style: Options in Composition (now out of print), Winston Weathers made a strong case for going beyond strict definitions of correctness when teaching style. Students should be exposed to a wide range of styles, he argued, including the "variegated, discontinuous, fragmented" forms used to great effect by Coetzee, Dickens, Mencken, and countless other writers.

Finally, thanks to @Cerberus, under the same relaxed rules, the figure can be considered asyndeton, from wikipedia example

"We must... hold them, as we hold the rest of mankind, Enemies in War, in Peace Friends."

Finally, the reason why I want to be tolerant of the fullstops is that I find it interesting and rewarding to interpret them as commas, but with certain dramatic pause of undetermined length and tend to read the text as an internal monologue of the storyteller.

_{For answer on how common is this (I assume in literature?) head to literature.}

Solution 2:

How common is this style?

Two examples spring to mind.

The shipping news is almost entirely written in that style. Concerns a newspaper writer, inner thoughts composed as headlines, compressed narratives. Heart is broken, re-mended. Sea rolling in the background. Beautiful, poetic, spare.

James Elroy, literary crime writer, has taken this style as far it will go. It’s a lot of fun, but can give some people a headache.

The figure if speech is 'ellipsis', but I don't know a name for it as an extended writing style. Saying something is 'elliptical' though, is usually a criticism of its intelligibility

Hmm. On this site it's hard to let this pass:

Omitting the subject from a sentence isn't proper construction, strictly speaking

Proper constructions are generated by grammar. Grammar is the bunch of rules you have in your head which allows you to take a finite set of words and construct completely new sentances which will be understood by other people. When you are a kid learning these rules, you often get it wrong, so you get taught Chapter 1 of these rules in school. Chapter 1 is a very small subset - probably what you are referring to as 'Proper Constructions'. The other chapters really only interest linguists, hobbyists, and more recently, computer scientists so most people never read them. These chapters will cover dropping subjects etc and such constructions are just as 'proper' as the ones they teach at school.

Solution 3:

When it comes to creative writing, be it fiction or non-fiction, proper construction goes out the window, and an author's writing style -- or the style of writing he or she chooses to adopt for a particular piece -- comes to the fore.

As this article on writing style notes, not only do styles vary by author, but particular styles of writing may fall out of favor. For example, notes the article, modern readers prefer Ernest Heminway's "concise, staccato" sentences to the florid prose popular in the Victorian era, as exemplified by George Bulwer-Lytton:

It was a dark and stormy night; the rain fell in torrents--except at occasional intervals, when it was checked by a violent gust of wind which swept up the streets (for it is in London that our scene lies), rattling along the housetops, and fiercely agitating the scanty flame of the lamps that struggled against the darkness.

The same passage, in Hemingway's style:

After dark a storm came and sometimes in the wind there was a noise on the rooftops. You could see the streetlamps struggling to stay lit.

This article credits Hemingway with pioneering this clear, brief style:

Hemingway pioneered a new style of writing that is almost commonplace today. He did away with all the florid prose of the 19th century Victorian era and replaced it with a lean, clear prose based on action rather than reflection... It is hard to find anyone writing today who doesn't owe a debt of influence to Hemingway.

So, to answer the OP's questions:

• This style of writing is more common in the modern era, especially post-Hemingway.
• It seems fairly common to characterize this style of writing as staccato. (Just Google staccato writing style.)