Of particular interest are the prayers and the order of reading the psalms in Nusach Ashkenaz.
So in the Nitei Gavriel on Aveilus (cheilek alef), chapt 10 it lists the halachos of those doing shemira. The following points are pertinent to your question:
(ois alef) One who is doing Shemirah is exempt from krias Shemah and any tefillah. (Refer to SA OC 71:3, SA YD 341:6 and SA YD 403:9). Therefore one can arrange, if possible, that there be two shomrim, so that one can guard whilst the other goes and says Shemah and davens. This idea I also saw brought down in the first cheilek of Gesher HaChaim, perek hei.
(ois ches) Even though it is forbidden to to daven and say divrei Torah in front of the deceased, it is permitted to say tehillim. (This idea is also brought in Gesher HaChaim linked above). Even an onein can say tehillim (this is further elaborated later here (ois gimmel)) as they are doing it to respect the deceased. However others are stringent since an onein by definition is patur (exempt) from doing any mitzvos.
So to answer your questions. General tefillos are avoided unless there is someone else there with you so that you can go and daven elsewhere whilst they guard. And as far as tehillim, one can recite (and I have read elsewhere that it is encouraged) but there is no set standard or formal selection.
However there is sefer which is cited by both the Nitei Gavriel and Gesher HaChaim entitled Ma'avar Yabok by Rabbi Aharon Berachia of Modina which provides select sections from throughout tanach (incl. Tehillim) that one can work through.
You did not answer my question. I asked about specific chapters of Tehillim and prayers, but for some reason you told me about halachot of shemira. Jun 8, 2022 at 19:37
The first two bullet points spell out the important background knowledge that one is exempt from prayers but one can say tehillim. and the third paragraph then answers the point.... Jun 8, 2022 at 20:37
Your question asked "What prayers and psalms are read during shmira" - firstly I proved that no "prayers" are said. Jun 8, 2022 at 20:37
What chapters of Teilim are read during shemira, according to Nusach Ashkenaz? I didn't find any information available! Jun 9, 2022 at 7:59
Exactly, as per my point - "one can recite (and I have read elsewhere that it is encouraged) but there is no set standard or formal selection" Jun 9, 2022 at 8:13
The ten Tehillim of Tikkun haKlali (16 32 41 42 59, 77 90 105 137 150, to be read in that sequence that they occur) are always appropriate and especially at Shemira. It is "The General Remedy." (As explained in Likutei Moharan #29 and other places).