The title of this dissertation encompasses the study of two disparate topics that have been worked on. All the results that have been obtained in this dissertation, as the fruit of three years of tedious work, are related to the following fields within Mathematical Analysis: • Algebraic genericity and lineability: This is the study of the algebraic structure within certain sets in a linear space or an algebra. In this sense, we study lineability and algebrability problems of sequences spaces and series. Just as, for the class of real singular functions on the unit interval. This topich as shown to be extremely fruitful in the last decade and this resulted in the American Mathematical Society introducing references 15A03 : Vector spaces, linear dependence, rank, lineability.46B87 : Lineability in functional analysis.in its latest Mathematical Subject Classification 2020...