Evaluate the expressions: $5^0,0!,1^5,(-1{)}^{2n}$ (where $n$ is an integer). Then list their numerical values in increasing order. If a value appears more than once, include it as many times as it occurs.

Let's evaluate each mathematical expression one by one:
1. $5^0$: Any non-zero number raised to the power of $0$ is equal to $1$. So, $5^0=1$.
2. $0!$: By definition, the factorial of $0$ is $1$. So, $0!=1$.
3. $1^5$: Any number $1$ raised to any finite power is $1$. So, $1^5=1$.
4. $(-1{)}^{2n}$: Since $n$ is an integer, $2n$ will always be an even integer. Any negative number raised to an even power becomes positive. Therefore, $(-1{)}^{\text{even}}=1$.

All four expressions yield a numerical value of $1$.
Arranging them in increasing order gives: $1,1,1,1$.