I want to find a $f \in L^p([0,1])$ where $p \in [1,2)$ such that the laws of $W$ and $W + \int^{\cdot}_{0}f_s ds$ are mutually singular. I know the following result where $D[0,1] = \lbrace F \in C([0,1]) \hspace{1mm} \mid \hspace{1mm} \exists f \in L^2([0,1]) : F(t) = \int^{t}_{0}f_sds, \hspace{1mm} \forall t \in [0,1] \rbrace$ and $(W_t)_{t \in [0,1]}$ is a standard brownian motion:
Let $F \in C[0,1]$ satisfy $F(0) = 0$. If $F \not \in D[0,1]$ then the laws of $W + F$ and $W$ are mutually singular.
Now for $p \in [1,2)$ my idea is to choose $b \in L^p([0,1]) \setminus L^2([0,1])$ (e.g. $b_s := \frac{1}{\sqrt{s}}$) and set $F(t) = \int^{t}_{0}b_s ds$. Then $F(0) = 0$ and $F \in C([0,1])$. Also if there exists $\tilde{b} \in L^2([0,1])$ such that $F(t) = \int^{t}_{0}\tilde{b}_s ds$ for all $t \in [0,1]$ then $b_s = \tilde{b}_s$ almost everywhere on $[0,1]$ and we would have $b \in L^2([0,1])$ which leads to a contradiction. Therefore $F \not\in D[0,1]$ and by the result from above we know that the laws of $W$ and $W+F$ are mutually singular.
Following the exercise 1.18 from Brownian Motion Book by Peter Mörters and Yuval Peres, we just need to find a continuous $F\notin D[0,1]$ so that $B+F$ has infinite quadratic variation.
From A uniform continuous function which is not Hölder continuous function $$F(t)=\cases{{1\over \ln (t/2)},&$0<t\le 1$\cr \strut0, &$x=0$}$$
is uniformly continuous but not $\alpha$-Hölder continuous for any $\alpha>0$ (and thus not $\frac{1}{\alpha}$-variation). And by differentiating we have
$$F(t)=\int \frac{1}{s\ln^{2}(s/2)}ds,$$
where $(s\ln^{2}(s/2))^{-1}\in L^{1}[0,1]$ but not in $L^{p}[0,1]$ for $p>1$. You can also verify by-hand that for $t_{i}:=\frac{i}{n}$
$$\sum |\frac{1}{ln(t_{i}/2)}-\frac{1}{ln(t_{i-1}/2)}|^{2},$$
goes to infinity as $n\to +\infty$.