Is the set that contains the empty set {∅} also a subset of all sets? I think it isn't; although $\emptyset$ is a subset of every set, $\{\emptyset\}$ is only a subset of a set $A$ if $A$ contains the element $\emptyset$. So if $A = \{\emptyset\}$, then both $\emptyset$ and $\{\emptyset\}$ are subsets of A, but if we have a set $B$ where $B = \{0, 1\}$ then only $\emptyset$ is a subset of it.
If my reasoning is true, then if there was a set $C$ where $C = \{\emptyset, 0, 1\}$, would $\{\emptyset\}$ then be a subset of $C$?
Moreover, is this a valid set (i.e. can you have the empty set as an element of a set that contains other elements)?
 A: Not only can you have the empty set as a member of other sets, but the classical way of building the natural numbers out of set theory does nothing but nest empty sets inside other sets.  We do that by defining $0=\emptyset$,  then $1=\{0\}$,   and in general,  $n=\{0,1,2,...,n-1\}$.   Writing these out long form gets you tons of empty sets!
A: Your reasoning is right. If $A=\{ \emptyset \}$ then both $\emptyset, \{ \emptyset \}$ will be subsets of $A$.
Applying the same reasoning, if $C=\{ \emptyset, 0,1 \}$ as @oliverjones states we can say $\{1\} \subset C$ because we have a $1$ inside $C$ in its definition (the same way there it’s an $\emptyset$ inside $A$ in its definition). With this in mind it’s easy to see that in fact $\{\emptyset\} \subset C$ and it’s a valid set.
A: I think, you have confusion between begin  element and begin a subset. Consider, the following example $$A=\{\emptyset, \{\emptyset\}\}$$ Now, $\emptyset\in A$ and $\{\emptyset\}\in A$. But, $\{\{\emptyset\}\}\subset A.$ I hope that helps.
A: To answer the question in the title, simply use the definition of "subsets".

$A$ is a subset of $B$ if and only if "every element of $A$ is an element of $B$".

So your question is equivalent to

is $\emptyset$ an element of an arbitrary set?

Of course not: the empty set is a subset of any set, but it is not necessarily an element of a set. The set $\{1\}$ is such an example.
