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In the projective planes $\mathrm{PG}(2,q)$, we collect the smallest knownsizes of complete arcs for the regions \begin{align*} &\mbox{all }q\le160001,~~ q \mbox{ prime power};\\ &Q_{4}=\{34 \mbox{ sporadic }q'\mbox{sin the interval }[160801\ldots430007], \mbox{ see Table 3}\}. \end{align*}  For $q\le160001$, the collection of arc sizes is complete in the sense thatarcs for all prime powers are considered. This proves new upper bounds on thesmallest size $t_{2}(2,q)$ of a complete arc in $\mathrm{PG}(2,q)$, inparticular \begin{align*} t_{2}(2,q)&<0.998\sqrt{3q\ln q}<1.729\sqrt{q\lnq}&\mbox{ for }&&7&\le q\le160001;~~(1) \\t_{2}(2,q)&<\sqrt{q}\ln^{0.7295}q&\mbox{ for }&&109&\le q\le160001;~~(2)\\t_{2}(2,q)&<\sqrt{q}\ln^{c_{up}(q)}q,~~c_{up}(q)=\frac{0.27}{\ln q}+0.7,&\mbox{for }&&19&\le q\le160001;~~(3)\\t_{2}(2,q)&<0.6\sqrt{q}\ln^{\varphi_{up}(q;0.6)}q,~~\varphi_{up}(q;0.6)=\frac{1.5}{\ln q}+0.802,&\mbox{ for }&&19&\leq\le160001.~~(4) \end{align*} Moreover, the bounds (2) -- (4) hold also for$q\in Q_{4}$. Also, \begin{align*} t_{2}(2,q)&<1.006\sqrt{3q\lnq}<1.743\sqrt{q\ln q}&\mbox{ for }&&q\in Q_{4}.~~(5) \end{align*} Ourinvestigations and results allow to conjecture that the bounds (2) -- (5) holdfor all $q\geq109$.

Tables, bounds and graphics of the smallest known sizes of complete arcs  in the plane $\mathrm{PG}(2,q)$ for all $q\le160001$ and sporadic $q$ in the  interval $[160801\ldots 430007]$