I might be wrong since I'm unable to find any sources supporting this, but it's increasingly my gut feeling that linguistics appears to focus on spoken languages as opposed to written ones. If this is the case, why is it?
Linguistics is the scientific study of language. A language is narrowly defined as the set of rules that "speakers" (speaking or signing) acquire when they are very, very young. There is evidence for processes of language acquisition underway at the very youngest testable ages (under a year old).
A speaker with a grammar like mine knows, without ever being told, that strings of words like colorless green ideas sleep furiously are well-formed, but strings like furiously sleep ideas green colorless are not. I can say that the first sentence would be true if (i) there existed some things that were colorless, green, and ideas, (ii) such things could sleep, and (iii) such sleepings could happen furiously. The other string supports no such interpretation. Since I don't have to look up these two strings in some kind of book to tell whether one is good or bad, we say that my grammatical rules are internalized.
The rules of grammar (again, in the linguist's sense) are learned implicitly. They are not taught, as the rule "i before e except after c" has to be learned when one learns how to spell, but rather inferred on the basis of experience in a linguistic environment (i.e., a community of speakers who do some speaking in the presence of the child).
Lastly, knowledge of language is universal. Barring severe cognitive deficits or social deprivation, every normally-developing child acquires a native tongue.
Written language stands in sharp contrast: children learn to read and write much later (usually once they start attending school) than they learn to understand and produce spoken language; written language has no internalized rules, it reflects only the speakers' internal rules and, possibly, a speaker's regard for the prescriptive rules of style; written language has to be learned explicitly (often laboriously); and, written language is not universal—there are many, many people on this planet who have knowledge of language but can't read or write.
"Written language" is an artifact of (some) human cultures who already had spoken languages. Spoken language is definitional of our species, and it is this kind of species-specific capacity that linguists study. This is not to say that the capacity for writing is not complex; it is in fact more complex than the capacity for language, in that explaining how we can have writing systems requires in part an explanation of what the language we're writing down is in the first place.
Your question is ambiguous, so I'll just cast my net wide.
There is a lot of research into different aspects of language that don't necessarily involve speaking. Psycho- and neurolinguistics, both thriving subdisciplines, focus their efforts on understanding the psychological and neurological components of language production and processing. Morphology, phonology and syntax focus on the rules that govern the production of language, not necessarily the particular modality of production. A lot of language acquisition research, such as that focusing on how babies process pointing gestures or gazes, or on the strategies that children use to learn a language, has very little to do with spoken language.
A lot of disciplines that prima facia deal with spoken language, such as phonetics or phonology, can and have been successfully adapted to sign languages. For example: research has shown that their are sign language analogues for place and manner of articulation, and that sign language has units of production that can be rightfully refereed to as phonemes.
Written language is a reflection of spoken, so most of what we know about spoken language can be transfered to written. The converse is not true. Still, there is a great deal of research focusing on how written language is processed.
Spoken language is a part of what makes us human. Every child will become fluent in their native language unless they suffer from massive cognitive or social deficits. Even mentally retarded children, or others with low IQ's, still achieve a great deal of fluency. If someone is deaf or mute, they will either acquire or invent a signed language. Acquiring literacy, on the other hand, is laborious process. Most languages do not have a written form, and even in those that do, not everyone achieves literacy. Spoken language is more widespread and universal, and therefore more interesting than the mostly cultural invention we call 'writing'.
Finally, a lot of linguistics has to be done with spoken language. A lot of phonetics, for instance, is about sound production, transmission, and perception. Children learn spoken language far earlier than written language, so most interesting research on language acquisition is about spoken or signed language. As I pointed out above, most languages do not have a written form, so any linguistic research concerning them has to focus on their spoken forms.
The attitude that spoken language is somehow primary to written (usually in a more general sense than geneologically) is known as logocentrism and has been extensively studied by, amongst others but arguably most famously, Jaques Derrida in his Of Grammatology.
To make an analogy, spoken language is like a hand and written language is like a hammer.
We are born with the capability to speak/hear (specific anatomical structures in the ear, mouth/throat/lungs, and brain) and we spontaneously learn spoken language by its presence. But writing systems are artificial, man-made contrivances, a coding system to record the spoken language, a tool that attempts to extend the usefulness of spoken language.
One naturally learns language growing up around people, but one needs instruction in reading and writing to do it. People communicated for thousands of years quite well without writing. The rules of writing are totally dependent on those of language that came before it.
To understand the arbitrariness of writing consider that Mandarin is normally expressed in logograms (the complicated picture-like collection of strokes for one syllable), but can also be expressed in pinyin, a romanization, roughly one letter per phoneme. Similarly, Persian is normally written in an Arabic script (leaving out most vowels) where Arabic is a very different structured language, but before the Arabic invasion was written in cuneiform, a syllabic system, but can be transcribed to a roman lettering. All these forms are with no change in meaning. The point is that writing systems are arbitrary inventions.
Writing is merely an attempt at encoding natural language (the non-written, primary, spoken thing). So, studying writing is studying the arbitrary man-made patterns of orthography, and these structures are very different from (or, if not, depend entirely on) the patterns of spoken language.
Studying writing (an interesting topic) is like studying a hammer, a very different thing than studying the hand that uses it.
As an aside, sign language needs to be accounted for (only because it is nominally not spoken), and that account (by analysis and experiment) seems to be that it has the same properties as spoken language in contrast to written ones.
For reference, see the preface/first chapter of any introductory linguistics textbook. There is the speech and writing section of wikipedia, which supports what I say, but you should seen actual authority to be convinced if you need a trustable reference.
It's pretty simple, actually.
Spoken language is a natural phenomenon; written language is a craft or art form.
There's a pretty strong divide here among the natural sciences. Chemists don't study Money; physicists don't study Beethoven.
Linguists don't study written language (generally) because written language isn't language — it's art.
Speech is thousands of years older than writing (the earliest known writing is Sumerian cuneiform -pictographs inscribed in clay). So it has primacy over the written word. According to Crystal (2003:178-9) both speech and writing are recognised as 'alternative, equal systems of linguistic expressions'. Focus on writing is characterised by studies in to literature and therefore style, genre and, perhaps, the notion of linguistic excellence and standards. Historically, speech was not considered worthy of study. In modern times, there has been much wider analysis of speech for a number of reasons: 1. Prevalence of 'second' language learning. 2. Investigations in to primary language acquisition. 3. Cognitive studies in to speech and the language centre of the brain. 4. Striving for improved standards and teaching methodologies. 5. etc..
or perhaps for the very fact that not a lot of academic attention had been paid to it before.
Speech is an innate skill that manifests itself without a formal learned environment (taking Krashen's distinction on 'acquisition' and 'learning'). Writing is a learned skill in formal learning settings. Children must learn to talk before they can write. So the primary focus in research is naturally on speech as the starting point of language acquisition.
Many linguists focus on the spoken language indeed, or at least they claim to. Often times, Ferdinand de Saussure is attributed with, among other things, steering linguistics into this direction around the turn of the 20th century. Alas, for many years to come, linguists saying they’re studying the spoken language were instead relying on written records thereof. This remains a problem today, although the methods of transcription have improved and modern technology makes spoken language better accessible to direct analysis. Yet still, it’s much easier to generate corpora from written sources. As a result, most books(!) on grammar describe literal grammar, not oral grammar, which can be very different – therefore many people (even some linguists) will regard perfectly well-formed oral utterances as “grammatically wrong” or “ungrammatical”. (Note the etymology of grammar.)
There’s the interesting notion of conceptually and medially oral and literal languages, prominently by Koch/Oesterreicher. That means, a book is written in concept and medium, like a face-to-face conversation is spoken in both, but a formal speech is conceptually literal and medially oral, whereas a chat message usually is conceptually oral and medially literal.
Of course, gesture/sign language is primary to orally spoken language which in turn is primary to written language. This is true both phylogenetically for humankind and ontogenetically for each individual (even though you can learn a second language in literal form). The fact that writing was created, hence is an artifact, doesn’t mean it’s mere recording of speech (nor thought, as some believe), though. The (conceptually) written form of language developed specific rules that don’t apply to the spoken modality. There are also several things that one can only do in writing proper, although Ong, Watt and many more have provided interesting examples of what spoken language is capable of (especially in non-literate societies).
Many languages, most probably, aren’t written at all, i.e. they may be transcribed sometimes, mostly by linguists, priests and administrators, but conversation and “literature” is oral/aural only. That doesn’t require the society to be illiterate at all, since written communication is then often done in a different language. For these, it doesn’t make much sense to study their writing separately, but for most European, many Asian and some other contemporary languages, their writing system (incl. orthography and script) and written grammar are independent objects of linguistic study.
As a student of mostly dead "classical" languages, most of the "linguistics" I did encounter, was and is historic linguistics aiming at a reconstruction of a postulated "proto"-language, predating the known and handed down languages, which again predate the spoken languages of today.
Now, while from my perspective of a semi- to quarter-linguist it makes perfect sense, that linguistics should focus on language as opposed to literature (or: "art", as esoneill put it), already the question struck me as quite absurd (regarding my experience), since the linguistics I know does not only not focus on spoken language but even near-totally suppresses any reference to "language" as something spoken rather than written. This is, of course, because of the fact, that written evidence endures much longer than the sound of an utterance and so is the only available data for this sort of historical linguistics.
Still, sometimes even here, stress is laid on the spoken languages. Of course not so much in what concerns "proto"-languages, but there are, for example, attempts to reconstruct the phonology of (Vedic) Sanskrit and it is quite well known that Classical Sanskrit never was a spoken language at all. (Incidentally, this holds true for most of the "classical" Indian languages). This kind of reverses the modern approach to languages, reflected in the answers already given: spoken language is here only appended and ancillary to the written.