Looking for some real world examples for mode in Statistics involving topics which students like say

Note that for the mean of a dataset to make sense, it needs to make sense for you to add together elements of the dataset and divide them by a scalar quantity. Essentially this means that the values must be elements of a vector space over $\mathbb{Q}$ or $\mathbb{R}$ (rational numbers or real numbers), or at least embeddable into a real or rational vector space. Examples include datasets where the points are integers, real numbers, complex numbers or elements of ${\mathbb{R}}^n$.
For the median to make sense the data values don't need to be members of a vector space, but they do need to have a linear order defined (so that you can pick out the 'middle value'). An example where the median makes sense but the mean doesn't is if the dataset consists of grades from $A$ to $F$, where you have an order $A>B>C>D>E>F$ but it is meaningless to add together two grades. On the other hand, if each data point is an $(x,y)$ coordinate than you can find the mean, but there is no sensible ordering so you can't find the median.
The mode always makes sense, because all you need to do is pick the most frequently occuring value. In particular, it makes sense for nominal data where there is no ordering, and it doesn't make sense to add together different data points. For example, consider a survey of surnames in the UK. You can't add together two names and it doesn't make sense to order them (unless you introduce an artificial ordering, such as lexicographic ordering, but that would be meaningless) but you can perfectly well count the occurences of each name and pick the most frequent one.
For some data the makes more sense though. If your data points are numbers drawn randomly from the interval $[0,1]$ then it's extremely unlikely that any of them will be repeated, so there is no 'most frequent' value. In cases like this it is common to put the data into bins (e.g. $[0,0.1),[0.1,0.2)$ etc) and then calculate the mode of the bins.