Bsrgvty

I've seen a few simple sentences leave out an "ich" and go straight to the verb, particularly in very informal conversation or for stylistic reasons.
Here are some examples that come to mind:

In the usage "How's it going?" -- "it's going."

Läuft.

In a short, half-hearted introduction:

Bin [name].

In this line from a song, though I'm unsure if it's meant to be part of the sentence before it. The song is Wolke 4.

Bin zu tief gefallen.

This is quite common to do in Spanish, but I see it much less often in German. Is this a normal or common kind of colloquial speech in German, and in which cases would it be appropriate?

A number of languages allow omitting pronouns (even) in formal speech and writing. Spanish is one of them, although the pattern is restricted to subject pronouns.

German, like English, is generally "non-pronoun-dropping". However, in informal and colloquial speech, it's not uncommon in either language. Some often-heard examples:

Kann sein. (Could be.)

Weiß nicht. (Don't know.)

Mach ich. (Will do.)

Linguistically, these omissions are ellipses (Ellipsen). As such, they are not governed by a firm grammatical rule, but by speech register and style. The complete phrases would be:

Es kann [so] sein. (It could be [so].)

Ich weiß nicht. (I don't know.)

Das mache ich. (I will do that.)

In an informal context, it is perfectly normal and not unusual to leave out these pronouns. And although I cannot provide a general rule (as per the above), this is not just true for these examples, but for the majority of ordinary phrases where the sentence's leading subject or object can be easily inferred from context. In fact, Wikipedia, in the article linked first, observes (albeit without a reference):

Colloquial and dialectal German, unlike the standard language, are also partially pro-drop; they typically allow deletion of the subject pronoun in main clauses without inversion, but not otherwise.

In the question's second example, I would consider the "half-hearted" introduction borderline impolite, unless it was immediately preceded by other people introducing themselves using the full sentence ("Ich bin …"). Simply because its demonstrated brevity runs counter to common courtesy. The third example, the line from the song, is however neither informal nor colloquial in style. The elliptical usage there lends a decidedly poetic touch to the lyrics.

A recommended read on the topic is Vorfeld-Analepse bei Aussagen. It's rather long and difficult, but covers a lot of ground. Custom requires that I mention at least a few pertinent points.

• It is not wise to conceive of this as dropping pronouns. It's about the Vorfeld being left empty in declarative sentences. In contrast, in a pro-drop language such as Italian, the subject pronoun is optional in interrogatives as well: Capisci?



• 
  

  That analepsis is limited to the Vorfeld is shown by examples such as the following.

  
  
  

  
  

  Und was sagst du zu den neuen Plänen der Regierung?
  
  Kenn' ich noch gar nicht.
  
  *Ich kenn' noch gar nicht.

  
  

  
  


• 
  

  In an example like the following, the dropped element seems to be da, which is not a pronoun and itself refers to drauf, which stands in for a prepositional phrase.

  
  
  

  
  

  A: Was denkst du über das Fest?
  
  B: Freu' ich mich riesig drauf.

  
  

  
  


• 
  

  I find bin (der) David impossible and the following example seems to support this. The idea is that rhematic (new) information cannot be left out.

  
  
  

  
  

  Wer hat hier geraucht?
  
  *Hab' geraucht.

  
  

  
  


• 
  

  In contrast to pro-drop, these kind of sentences require context to work, although the context can be extralinguistic.

  
  
  

  
  

  Macht nichts [mit Verweis auf ein vorgefallenes Malheur]

  
  

  
  

This is very uncommon in German and I’m not even sure off the top of my head where it would be heard. While personal pronouns can be reduced a lot (the furthest would probably be Bavarian Wenn’st mogst, if you want, where the st on wenn is essentially what replaces du) they are practically always there in one way or another.

Word order is very flexible in German though, so a personal pronoun may not necessarily be anywhere close to the beginning of a sentence – although the soft ordering rules usually suggest they should be somewhere close-ish.

Edit to include your examples:

Läuft as an answer to ‘how are you’ has almost taken on a life of itself and it might be considered a fixed word rather than a full sentence.

Bin (der) Jan (Southerners like me would include an article, Northerners are more likely not to): Yes, possible, and there are a couple of other occasions where a short answer might be given without a personal pronoun but to me this case gives a vibe of ‘here’s your answer, I’ve been polite, now leave me in peace’. A similar example might be if you fell over and are asked if you’re fine you might answer ‘Geht schon wieder’ – the shortest possible reply. It’s not a very common thing and it only applies to rare cases.

As for music, poetry and literature: they follow their own rules and don’t necessarily form full sentences if it sounds better in another way.

This is called pro-drop and German is not called a pro-drop language. Quite the opposite, we even add a pronoun when it's not needed at all, Es ist mir klar, "Mir ist klar das" (Mir ist klar, dass "das" wird so nicht mehr verstanden), "Es regnet", where Spanish for example uses just a single word for the later.

Anyway, it would be deemed higly incorrect to say "* Will spielen!".

The exception is perhaps "Bin da, wer noch", as known for example from the lovely show Die Dinos, "Bin schon da", "Bin mir nicht sicher", etc. This could be explained as a special exception, if bin was a particularly strong verb, reflecting a preverbal root akin to "bei", "be-come". I'm not sure how this related to certain moods like the optative mood (na los, lass losgehen "let's go"). The imperative mood does not require a pronoun, still. All Indo-European languages have lost the optative (except rudimentary fragments in Greek), supplemented or conflated with other moods. Cp. ich wäre dann be-reit, I [am] be ready; regularly bei'm [gerundive] Gehen, * ich bin gehen "I am going"; Bei Fuß! (a commando given to dogs). Also cp. "wird gemacht", Eng. "will do".

Do examples of other such words exist? Könnte sein!