Ich bin angefangen is a grammatical variation that does not exist in standard German. It is generally considered wrong, and even those who use it colloquially seem to be fully aware of its non-standard nature, so that they normally ‘correct’ it when writing. There are not enough occurrences of this variation in print for Google’s n-gram viewer to have any data on it. The few hits for it in Google Books also support this. Some are false positives such as “… dass ich … bin. Angefangen mit …” or OCR errors. Many are extremely literal translations from Latin in Latin courses. Many of the remaining ones set up a specific non-standard context or discuss the variation, rather than just use it. For example:
Ich übte die Hamburger Ausdrücke, ich sagte „Feudel“ statt „Lappen“ und „ich bin angefangen“ statt „ich habe angefangen“.
In colloquial North German, anfangen and beginnen form their perfect with sein. One thus hears ich bin angefangen, begonnen for standard German ich habe angefangen, begonnen.
Außerdem muss man sich bewusst machen, daß auch die ungesteuerte Herausbildung von Handlungsnormen immer auf einem ‚aristokratischen Modell‘ beruht. // Betrachten wir in diesem Zusammenhang einmal die folgenden Formulierungen: // (1) Im Deutschen heißt es nicht ‚ich bin angefangen‘ sondern ‚ich habe angefangen‘. […] (3) Heute sagt man auch ‚zu Beginn diesen Jahres‘ statt ‚zu Beginn dieses Jahres‘.
Und in einigen Sprachgebieten Nord- und Westdeutschlands sagt man bin – angefangen (aber immer: habe – begonnen).
Der Standard wird auch oft mit der Norm der Sprache schlechthin gleichgesetzt. Die Aussage „Das ist normwidrig im Deutschen“ meint dann einfach: ‚Es ist kein Standarddeutsch‘, z.B. das sein-Perfekt bei anfangen: Ich bin angefangen usw. Sprachwissenschaftler sind natürlich klüger und wissen, dass auch Nonstandardvarietäten Normen haben, dass man z.B. fehlerhaft Dialekt sprechen kann.
What remains is some extremely literal reports in which the traditional German editing is not done, sometimes apparently due to the application of Anglo-Saxon, more ‘scientific’ editing principles. Examples:
Ich bin angefangen mit einer Handpuppe, und die kommt dann immer wieder vor.
Ich bin angefangen, kurz nachdem ich aus der Gefangenschaft kam, mit einem sehr netten Gebrauchtwagen.
Ich bin angefangen mit 950 DM.
Ich bin angefangen als ehm [sic, no OCR error] stilles Mitglied und habe mich sehr schnell abgeschreckt gefühlt von den ganzen Abteilungsarbeiten.
Ich bin angefangen im Zuchthaus.
Ich bin angefangen mit dem alten Thoreau und hab mich von dem seitwärts in die Büsche geschlagen, die der amerikanischen Literatur.
Nun gut, ich bin angefangen als Schiffsjunge für die Hanse, deren Schiffe damals London häufig angefahren haben.
Most sources discussing the variation point out that it is northern or (north-)western. Some locate it in North-Rhine Westfalia, others in Hamburg. A similar phenomenon appears to exist in Dutch. The officially correct variant is ik ben aangevangen, but a Dutch dictionary from 1829 (Bilderdijk’s Nederduitsche Spelling) also lists ik heb aangevangen without comment. But it is not, after all, exactly the same phenomenon. First, it seems to be a lot more common in Dutch. Perhaps more importantly (and explaining the different frequency at least in part), as you can see in the examples above, there seems to be a strong tendency for Germans who use bin angefangen to only prefer it when starting their own life story. This tendency, which I have not found pointed out in any of the sources discussing the variation, is not apparent in the Dutch examples.
Rather surprisingly, I have also found examples of this variant conjugation in an English course from 1854 printed in Munich:
I have begun to sing. Ich habe (bin) angefangen zu singen.
I had begun to drink. Ich hatte (war) angefangen zu trinken.
This is probably a non-native speaker’s mistake, as it appears that the author was English.
There is a map for the modern distribution of this variant over at Atlas zur deutschen Alltagssprache: ist/hat angefangen. The Atlas notes:
Hier ist ein Einfluss des niederl. Verbs beginnen (‘beginnen, anfangen’), das mit zijn (‘sein’) gebildet wird, denkbar.
The general context of this variation is as follows: The composite past started with be for some verbs and with have for some other verbs, depending on what initially made more sense. Then it became a purely grammatical construction and generalised to the point where you could use one or the other variant for every verb. Currently we seem to be in a process of regularisation in which overall the remaining cases of sein are slowly being replaced by the more common haben. (In English this has already completed. In Shakespeare’s time one could still say “I am gone to the theatre”.) As a relatively stable intermediate stage, we currently have a system where, similar to French, verbs of location and movement are conjugated with sein and everything else with haben. But just like the German and French interpretations of this principle differ in the details, so do the northern and southern German interpretations. In the south, the general idea seems to be that it is a property of every verb whether it is conjugated with sein or with haben, in the sense that e.g. sitzen can only be correctly conjugated one way or the other, regardless of how it’s used. In the north, the general idea seems to be that by conjugating the same verb either with sein or with haben you can express subtle nuances such as direction (sein) or mode (haben) of a movement. I guess something similar is the root of the modern Dutch distinction between beginnen (conjugated with zijn — more abstract?) and aanvangen (conjugated with hebben — more concrete?).
It appears that bin angefangen was once a relatively common variant, which makes sense for semantic reasons. (Though it never seems to have been part of standard German, as it is practically absent even in old sources.) In the south and east it has disappeared completely, but in the north and west it has survived — especially in those contexts where anfangen is used to describe the start of a career.