While the two cases you give seem to depend on the same issues, there are halachic differences between them, which are also affected by timing. We will start with the case of wax.
Wax is not a kli and is not normally fit for use on Shabbat, as burning it is forbidden. It is thus muktzeh machamat gufo, the basic level of muktzeh. If wax started dripping onto the plate before Shabbat and one’s intention when placing the plate was to catch the wax, the plate could theoretically become a bassis l’davar ha’asur (an otherwise permitted object that becomes muktzeh by serving as a base for something muktzeh). However, it is likely that droplets of wax are not important enough to affect the status of the plate in that case.[1] (In contrast, a candlestick resting on the plate would likely make the plate a bassis l’davar ha’asur.[2])
The matter is even more lenient if one did not put the plate under the candle until after certain nightfall of Shabbat (after tzeit hakochavim[3]), as a bassis l’davar ha’asur cannot be created in the midst of Shabbat.[4] In such a case, if one wants the base utensil, he can shake off the muktzeh item and use the utensil. Furthermore, if it is not possible to remove the muktzeh, or even if it would just cause damage to do so, one can use the base as is.[5] The problem, however, is that it is forbidden on Shabbat to take a kli out of use (mevatel kli meihechano). According to some, this applies even if you could remove the muktzeh and thereby return the plate to use.[6] For this reason, the Shulchan Aruch[7] forbids putting a kli under a candle to catch oil. The Mishna Berura[8] says that this can be remedied by putting something usable on Shabbat on the kli before placing it under the candle, whereby the kli can still be moved.
Now we will discuss your example of water. If one placed a bucket before Shabbat to catch the water, then the situation depends on the state of the water. If the water is fit for washing or animal consumption, there is no problem of muktzeh.[9] If the water is not usable, it is muktzeh, and the bucket is therefore a bassis l’davar ha’asur. However, the gemara[10] says that just as a g’raf shel re’i (portable toilet) can be removed with the excrement it contains if it is in a place where people go about activities and find its presence disturbing, we can similarly apply this leniency to cases of milder disturbances. This includes utensils containing food residue,[11] as well as our case of a bucket filled with unclean water if it is found in a room in use.[12]
Placing the bucket on Shabbat to catch the water[13] or returning it after spilling out the water is more problematic due to the convergence of two concepts. On Shabbat, one is not allowed to intentionally create a situation of g’raf shel re’i with the plan to move it.[14] Thus, one should decide when he puts the bucket under the water that he will leave it there. But if he does that, then he will have violated the prohibition of mevatel kli meihechano. The Tur[15] argues against this stringency, and the Bi’ur Halacha[16] suggests a proof that we rely on the Tur’s lenient opinion from the fact that people collect mayim acharonim and morning netillat yadayim water in a receptacle. (It is possible to distinguish between those cases and the case in question.[17]) Also, one can create a situation of g’raf shel re’i on Shabbat in order to avoid significant loss.[18]
If the water has already caused a g’raf shel re’i situation on the floor, one can remove the unseemly muktzeh even by hand, and certainly with or in a kli. Thus, one could sweep the water into a kli of some sort in the process of removing it.[19]