Subgraph isomorphism in planar graphs
I'm not aware of any implementations of planar subgraph isomorphism algorithms, sorry. Note that "SubGemini", which is a (1993) circuit/netlist-oriented subgraph isomorphism solver, doesn't use a planar algorithm, seemingly because they did not want to make planarity assumptions.

For subgraph isomorphism in general (i.e. not planar), the practical state of the art seems to be:


Lee, J. et al. (2012) "An In-depth Comparison of Subgraph Isomorphism
Algorithms in Graph Databases". The paper says they've released source code for their [re]implementations, but I'm having trouble find it... so you may want to email the corresponding author.
Solnon, C. (2010): "AllDifferent-based Filtering for Subgraph Isomorphism." Its implementation is found in LAD. 
Cordella, L.P et al. (2004) "A (sub)graph isomorphism algorithm for matching large graphs"; this is basically describing the implementation in vflib and in NetworkX.


All of these papers have experimental results, so you can get an idea what problem sizes they can handle (which might not be a bad idea to specify in your question).

The paper by Dorn (2009) suggested by Luke Mathieson above seems your best bet if you want to implement planar subgraph isomorphism, but that paper has no experimental results, so... you'll probably be the first to experiment with it.



Alternatively, since we're talking subcircuit extraction (SCE) here, you could look at  Jiang and Bunke's Marked Subgraph Isomorphism approach for SCE. The assumption there is different than planarity:


  In this case we know a priori that the degree of some vertices is preserved under the subgraph isomorphism mapping. This information is explicitly utilized to develop a polynomial-time algorithm.


I'm a bit skeptical about the applicability of that paper to real-life SCE because the paper was published at somewhat obscure workshop that is not even circuit-focused... But they claim to have soundly beaten SubGemini. The subgraphs searched there are pretty small, a 6 node cell (presumably 6T SRAM cell) is searched and found (in many copies) in a RAM array of several thousand copies. If your application looks like this... it might be what you want.

The 2003 IEEE Potentials article about SCE even though co-authored by Frank Harary alas ignores this paper of Jiang and Bunke entirely, which not really a surprise to me given where it was published. You may want to read that brief for pointers to other approaches to SCE, including genetic algorithms, neural networks etc. A couple of SCE papers indicated in that brief as claiming good practical results are:


Chang et al.'s DECIDE "A novel subcircuit extraction algorithm by recursive identification scheme"
Zhang and Wunsch "A fuzzy attributed graph approach to subcircuit extraction problem".


Not covered in that review because it's slightly newer is FROSTY; it works by matching smaller directed (sub)graphs by doing a first pass identifying gates. In its paper, results are only compared to SubGemini though, which isn't that hard to beat.

Somewhat similar (in the sense that it relies on higher-level graphs that are in a certain sense directed) but broader in scope (in the sense that it handles both exact and approximate matching) is "Pattern Search in Hierarchical High-Level Designs" by Terem et al. from Intel + Moshe Vardi. The way they handle both exact and approximate matching is setting up both problems a constraint satisfaction problems.

Also probably worth looking at is a series papers by N. Rubanov 2006; 2002; his methods are probabilistic, but they all beat SubGemini by orders of magnitude. Rubanov worked in the EDA industry (Circuit Semantics, Magma Design Automation, Cadence).

There also three fairly recent Oracle (prbably Sun labs) patents by Douglas C. Meserve in this area: US7958468,
US7861193,
US8788990. I couldn't find any academic publication related to these patents.

The most recent comprehensive SCE work I found (insofar) is a 2009 Ph.D. thesis "Hierarchical Pattern Matching in VLSI" by M. Miloševic. 

As an interesting addendum, SubGemini is based on the Conneil and Gotlieb partitioning, which has its issues, in these sense that its correctness is based on a conjecture that was eventually proved false by counterexample.