For all uniform classes $\operatorname{UF}$, for essentially any non-uniform class $\operatorname{NU}$, one will
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have $\: \operatorname{NU} \not\subseteq \operatorname{UF} \:$, $\:$ since $\operatorname{NU}$ should be uncountable and $\operatorname{UF}$ is countable.
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("Proof:" $\;\;\;$ By any reasonable definition of being uniform, for a language to be in $\operatorname{UF}$
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there must be something in a particular at-most-countable set of things such that the
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thing determines the language, so there are at most countably many languages in $\operatorname{UF}$.
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On the other hand, for each subset $S$ of $\:\{\hspace{-0.03 in}0,\hspace{-0.05 in}1,\hspace{-0.03 in}2,\hspace{-0.03 in}3,...\hspace{-0.04 in}\}\:$,$\:$ the definition of $\operatorname{NU}$ should
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mean that [the language consisting of exactly the strings whose length is in $S\hspace{.02 in}$] is in $\operatorname{NU}$.
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In that case, since those languages are all different, $\operatorname{UF}$ will be uncountable.
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"Therefore," one will have $\: \operatorname{NU} \not\subseteq \operatorname{UF} \;$.)

In particular, $\;\;\; \mathsf{non-uniformAC^0} \: \not\subseteq \: \operatorname{PSPACE} \;\;\;$ and
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$\mathsf{non-uniformACC^0} \: \not\subseteq \: \operatorname{NEXP} \;\;\;$, $\;\;\;$ so both claims in your initial sentence are false.
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It is far more interesting to ask which uniform classes
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are subsets of which non-uniform classes.
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[$\operatorname{REG}$](https://complexityzoo.uwaterloo.ca/Complexity_Zoo:R#reg) $\: \not\subseteq \: \mathsf{non-uniformAC^0} \;\;\;$, $\;\;\;$ since $\;\;\;$ [parity $\: \not\in \: \mathsf{non-uniformAC^0}$](http://www.sciencedirect.com/science/article/pii/S0020019011003504) $\:\:\:\:$.
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$\operatorname{NEXP} \: \not\subseteq \: \mathsf{non-uniformACC^0} \;\;\;$; $\;\;\;$ which [your question on cstheory](https://cstheory.stackexchange.com/q/31290/6973) was asking about.
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[$\operatorname{MA_{exp}}$](https://complexityzoo.uwaterloo.ca/Complexity_Zoo:M#maexp) $\cap \hspace{.03 in}\operatorname{co-MA_{exp}}$ $\: \not\subseteq \:$ [$\operatorname{P/poly}$](https://complexityzoo.uwaterloo.ca/Complexity_Zoo:P#ppoly) $\;\;\;$, $\;\;\;$ by [page 16 of this paper](http://www.uni-ulm.de/fileadmin/website_uni_ulm/iui/Ulmer_Informatik_Berichte/1994/UIB_1994-11.pdf#page=16).
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[$\operatorname{AM_{exp}}$](https://complexityzoo.uwaterloo.ca/Complexity_Zoo:A#amexp) $\: \not\subseteq \:$ [$\operatorname{(NP\cap coNP)/poly}$](https://complexityzoo.uwaterloo.ca/Complexity_Zoo:N#npiconppoly) $\;\;\;$, $\;\;\;$ by [this paper](http://ac.els-cdn.com/S0020019003004460/1-s2.0-S0020019003004460-main.pdf?_tid=c3830bfe-ee11-11e4-a51b-00000aab0f02&acdnat=1430272259_b81c933f49fc08903b34cf740d7739f5). $\;\;\;\;\;$ (The right-hand-side of
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the last non-containment is likely smaller than the class one would think it means.)
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[This paper](http://link.springer.com/chapter/10.1007/3-540-48686-0_21) shows that [$\operatorname{MA_{exp}}$](https://complexityzoo.uwaterloo.ca/Complexity_Zoo:M#maexp) can't be decided by circuits with "half-exponential" size.
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