  We investigate three cases regarding asymptotic associate primes. First,
assume $ (A,\mathfrak{m}) $ is an excellent Cohen-Macaulay (CM) non-regular
local ring, and $ M = \operatorname{Syz}^A_1(L) $ for some maximal CM $ A
$-module $ L $ which is free on the punctured spectrum. Let $ I $ be a normal
ideal. In this case, we examine when $ \mathfrak{m} \notin
\operatorname{Ass}(M/I^nM) $ for all $ n \gg 0 $. We give sufficient evidence
to show that this occurs rarely. Next, assume that $ (A,\mathfrak{m}) $ is
excellent Gorenstein non-regular isolated singularity, and $ M $ is a CM $ A
$-module with $\operatorname{projdim}_A(M) = \infty $ and $ \dim(M) = \dim(A)
-1 $. Let $ I $ be a normal ideal with analytic spread $ l(I) < \dim(A) $. In
this case, we investigate when $\mathfrak{m} \notin \operatorname{Ass}
\operatorname{Tor}^A_1(M, A/I^n)$ for all $n \gg 0$. We give sufficient
evidence to show that this also occurs rarely. Finally, suppose $ A $ is a
local complete intersection ring. For finitely generated $ A $-modules $ M $
and $ N $, we show that if $ \operatorname{Tor}_i^A(M, N) \neq 0 $ for some $ i
> \dim(A) $, then there exists a non-empty finite subset $ \mathcal{A} $ of $
\operatorname{Spec}(A) $ such that for every $ \mathfrak{p} \in \mathcal{A} $,
at least one of the following holds true: (i) $ \mathfrak{p} \in
\operatorname{Ass}\left( \operatorname{Tor}_{2i}^A(M, N) \right) $ for all $ i
\gg 0 $; (ii) $ \mathfrak{p} \in \operatorname{Ass}\left(
\operatorname{Tor}_{2i+1}^A(M, N) \right) $ for all $ i \gg 0 $. We also
analyze the asymptotic behaviour of $\operatorname{Tor}^A_i(M, A/I^n)$ for $i,n
\gg 0$ in the case when $I$ is principal or $I$ has a principal reduction
generated by a regular element.
