What should a person do if he is not sure whether he davened?

A person who is not sure whether he already davened should daven. Before he begins, he should stipulate the following: If I have already davened, this tefilla should be a nedavah (a voluntary offering), and if I have not yet davened, this tefilla should be the obligatory tefilla. Generally, tefillos nedavah are discouraged, since it is difficult to have the proper kavanah. In this case, however, most Achronim hold that since doing so is necessary in order to fulfill one’s (possible) obligation, he is encouraged--and according to some, obligated--to daven in this way. It is also insufficient just to daven a tefillas nedavah, since there is a possibility that he if is obligated to daven, his obligation cannot be discharged with a nedavah.

סי' ק"ז ס"א ס"ק ב' ביה"ל ד"ה אם וד"ה חוזר ביאורים ומוספים דרשו 2,3 

What should one do if he is not sure whether he davened on Shabbos?

Since tefilla was instituted as a replacement for korbonos, there is no option to daven multiple obligatory tefillos. However, since an individual may offer multiple korbonos nedavah, one who already davened may choose to daven additional tefillos nedavah.  A person may not offer a korbon nedavah on Shabbos and Yom Tov, however, so one may not daven a tefillas nedavah on these days. A Korbon Musaf is an obligatory korbon which is brought by the tzibur. As such, a person may not daven Musaf as a nedavah either. As we have learned, when a person is unsure whether he davened, he should daven with the  stipulation that his tefilla is a possible nedavah. On Shabbos and Yom Tov or during Musaf, when a nedavah is not an option, this method is not feasible. A possible alternative would be to serve in the role of chazan and daven Chazoras HaShatz. In this way, one will discharge his obligation in case he never davened.

ס"א ס"ק ג'-ו' ביאורים ומוספים דרשו 9

What is a prerequisite for a tefillas nedavah?

When one davens a tefillas nedavah, he must incorporate a new request. Doing so makes it clear that the tefilla is a nedavah and not a repeat of the chovah. The request must be incorporated into one of the middle brochos and must be a request for the fulfillment of a new need which he was not aware of when he davened (possibly) the previous time. In a case where a person is davening because he is unsure whether he previously davened, he does not need to incorporate a new request. Although he is stipulating that his tefilla is a possible nedavah, a new request is not necessary since the arrangement is being made is to fulfill a possible obligation.

ס"א-ס"ב ס"ק ב' ד' ח' 

• There is disagreement about whether a woman is obligated to daven Musaf. Some say that she is exempt, since it is a time-bound mitzvah and is not focused on requests for mercy.

• In the times of the Tanaim, there were people whose primary occupation was the study of Torah. Those people learned constantly and did not engage in earning a living (even for a moment). They stopped learning in order to fulfill most mitzvos, but they were exempt from the mitzvah of tefilla. Since tefilla is primarily a request for mercy, the requirement is not as stringent. As such, people in this category were not obligated to disturb their learning to daven.

• Nowadays, since there are no longer such people, everyone is obligated to daven--even if they are in the middle of learning.

• Are women obligated in tashlumin?

• When is one obligated to daven for a third time?

• Is it ever necessary to daven Shemoneh Esrei four times?