After about 1,800 answers that I have given here on German Stackexchange, this here is the first time I've made use of the option to post two answers to one question, because there is so much to say about this question. (In my other answer, I note that the claim that the preposition »wegen« is increasingly rarely used with the genitive is incorrect).
This answer here is generally about the use of the genitive and its role in language change.
I grew up on the outskirts of Graz in southeastern Austria (there is no L in Austria, so its not Australia!). My parents are from eastern Styria (mind the T in Styria, it's not Syria! Styria is one of the 9 states of Austria and Graz is the capital of Styria), and until I was six years old, all the people I came into contact with spoke the local dialect Oststeirisch (East Styrian), which is the local expression of the Middle Bavarian or Danube Bavarian dialects, with influences from neighboring regions. My grandfather, Christian Schölnast, even wrote books in this dialect. The special thing about this dialect is that it does not contain a genitive. (Dative and accusative are also merged into one case, with the exception of pronouns, but that is another topic).
There are no verbs in this dialect that govern the genitive (»gedenken«), there are no prepositions that require the genitive (»wegen«), there are no adverbial determiners in the genitive (»schnellen Schrittes«), and there are no genitive attributes (»des Orkans«):
Standard German: Der Vater eilte wegen des herannahenden Sturmes schnellen Schrittes herbei und gedachte dabei der anderen Kinder, die bereits früher ein Opfer des Orkans geworden waren.
East Styrian: Da Foda is wegn an Sturm, der daherkumman is, ziemli schnöö hergrennt und hod dabei an die ounan Kinna denkt, die scho friara Opfa vu dem Orkan woan san.
The father hurried at a fast pace because of the approaching storm, remembering the other children who had already been a victim of the hurricane earlier.
As far as I know, Oststeirisch is representative of all Bavarian dialects, which all get along quite well without the genitive.
I first learned the genitive in school, and at that time it seemed just as unnecessary and foreign to me as the new language to which it belonged: standard German. Today, as an adult, I see things differently, but my point is that there are already variants of the German language that can do without the genitive, although I observe that the number of speakers of Bavarian dialects is currently declining sharply, and I expect this group of dialects (or is it a language?) to become extint in the next 100 to 200 years. And the consequence of this will then be that, at least in Austria and Bavaria, the genitive will be used more frequently in the future than it is today.
But even without having to refer to the dying of the dialects, there is no sign of a demise of the genitive.
Prepositions with Genitive
The number of prepositions governing the genitive has increased, not decreased, in recent centuries. The words aufgrund, anstatt, infolge, anstelle and zugunsten have already attached the genitive in the past. Others, such as trotz, dank and laut have only recently moved away from the dative and are now used with the genitive (colloquially still often with the dative). And with the word entsprechend you can just observe live the transition from the dative to the genitive:
Dativ: Die Weiterverwendung dieser Daten ist entsprechend dem Informationsweiterverwendungsgesetz nicht zulässig.
Genitiv: Unsere Sicherheitsmaßnahmen werden entsprechend des technischen Fortschritts fortlaufend verbessert.
Interestingly, this transition seems to happen mainly in those regions where genitive-less dialects are spoken. In my research, I found examples of "entsprechend" + genitive only on Austrian websites (but there in large numbers). Of course, this could also have something to do with the fact that Google shows me Austrian pages preferentially, but I think it is more likely that in these regions the genitive is mentally linked with "elevated language", because it hardly plays a role in the ordinary everyday language. And because especially the dialect speakers in these regions generally find the genitive somewhat strange, they are also prepared to tolerate this strangeness in the elevated written language.
The genitive attribute is a suffixed attribute of a noun that can always be replaced by the also correct construction "von" + dative:
der Vater des Kindes, im Osten Österreichs
der Vater von dem Kind, im Osten von Österreich
But there are also cases where the genitive is not possible at all, namely when the attribute is plural and used without a determiner:
correct: Die Frage von Schülern, ob Masken getragen werden müssen, blieb unbeantwortet.
wrong: Die Frage Schüler, ob Masken getragen werden müssen, blieb unbeantwortet.
Apart from such exceptions, however, it is the case that in written German the genitive attribute is used almost exclusively, and a shift to the prepositional attribute is not recognizable.
In spoken everyday language - at least where I live - the construction "von" + dative has always been the preferred variant. However, as everyday language, especially in Germany and Austria, is becoming more and more like standard German due to the increased consumption of cross-national media (Internet, television), and as the number of dialect speakers is rapidly decreasing, it can be observed in spoken German that the construction "von" + dative is being pushed back more and more in favor of the genitive attribute.
When a verb rules the genitive, it is called a genitive object.
Dieser Vorfall hat ihn eines Besseren belehrt.
Karl befleißigt sich einer sorgsameren Sprache.
Der Täter wurde des Mordes beschuldigt.
Der Mann entledigte sich seiner Kleider.
Sandra unterzog sich freiwillig dieser Prozedur.
Walte deines Amtes!
About 150 to 200 years ago, there were many more verbs used with a genitive object. For many of them, the use of the genitive today is considered outdated or even wrong:
past: Egon trank und vergaß dabei des Auftrags. (Genitivobjekt)
present: Egon trank und vergaß dabei den Auftrag. (Akkusativobjekt)
past: Fürchtest du dich nicht der Sünde? (Genitivobjekt)
present: Fürchtest du dich nicht vor der Sünde? (Präpositionalobjekt)
past: Sie braucht des Geldes um Brot zu kaufen. (Genitivobjekt)
present: Sie braucht das Geld um Brot zu kaufen. (Akkusativobjekt)
Here we can clearly observe a decline of the genitive in the past. With the verb gedenken (»sie gedachten der Toten«) you can follow this change live right now. Still officially only the genitive is considered correct, but in some newspapers you can also find the dative:
Stuttgarter Zeitung: Der Niederösterreichische Landtag gedachte den Toten
Die Glocke (Nähe Münster, NRW): Die Bruderschaft gedachte den Toten aus Krieg, Terror und Gewalt
OVB Online (Rosenheim, Bayern): In einer Feierstunde gedachte die Pfarrgemeinde Oberbergkirchen den toten Soldaten der beiden Weltkriege sowie den Opfern von Terror und Gewalt am Kriegerdenkmal.
There is no verb that has taken a genitive adjective in recent centuries, but, as just shown, many that have lost it in the past. Whether "gedenken" will join this ranks is still uncertain.
In addition, however, there are a number of verbs which cannot be imagined without a genitive object, even at present. With these verbs the genitive seems to be very stably anchored, a change is not in sight here:
Der Zeuge beschuldigt August K. des Mordes an Liselotte R. Der Staatsanwalt klagt Herrn K. des Mordes an. Herr K entschlug sich der Aussage und sein Anwalt bediente sich einer List um seinen Mandanten zu entlasten, doch der Richter waltete seines Amtes und verurteilte Herrn K. des Mordes an Liselotte R. zu lebenslanger Haft. Liselottes Schwester erbarmte sich des hinterbliebenen Kindes und nahm sich seiner an. Später brüstete sich Herr K. sogar noch seiner Tat, doch entbehrte dies jeder Logik. Es zeigt aber, dass sich Herr K. seiner Tat nicht schämte.
The witness accuses August K. of the murder of Liselotte R. The prosecutor accuses Mr. K. of murder. Mr. K. refused to testify and his lawyer used a ruse to exonerate his client, but the judge used his authority and sentenced Mr. K. to life imprisonment for the murder of Liselotte. Liselotte's sister took pity on the bereaved child and took care of him. Later, Mr. K. even boasted of his deed, but this lacked any logic. It shows, however, that Mr. K. was not ashamed of his deed.
In conclusion, I would say that there has been a decline in verbs with genitive objects in the past, but that this decline has now bottomed out, and the proportion of genitive objects in written German will probably remain at about its present level for the foreseeable future. In spoken German, I expect more of an increase, for the same reasons I gave above.
Meines Wissens kam eines Abends Reinhard schnellen Schritts des Weges und erzählte Barbara frohen Mutes von seinen Plänen. Doch Barbara entgegnete ihm traurigen Herzens, dass ihnen eines Tages das Geld dafür ausgehen würde. Aber Reinhard entgegnete ruhigen Gewissens, dass ihre Reserven dafür ausreichen.
As far as I know, one evening Reinhard came walking quickly along the path and told Barbara about his plans in a cheerful mood. But Barbara replied with a sad heart that one day they would run out of money for it. But Reinhard replied with a calm conscience that their reserves were sufficient for it.
The adverbial genitive does not occur particularly frequently, but here too, neither an increase nor a decrease in frequency can be observed in written texts. For spoken German, the same applies as above.