It is hard to find a research topic. Here are some practical ideas to get you started.
(1) Asking a supervisor or staff member for some suggestions to get you started is a good idea. Usually professors want you to succeed and will assist with some general ideas. BUT: Professors hate it when you walk into their offices without ideas or not appearing to be prepared. They want to know that you are actively thinking and that you are not trying to just be lazy. So I advise that you go through each of the steps below and generate 3 or 4 possible kinds of studies. THEN take these ideas to your professor or tutor and see what they suggest.
(2) Try and avoid English: if you speak any other language, then consider doing a research topic in that language. English is a wonderful language, but it is very intensively studied and all textbooks are full of it. So why join the herd? Draw on other interesting languages that you speak or which are spoken around you!
(3) As a rule of thumb, if you are a new linguist, try and choose a topic involving fairly large structures (i.e. not just single words or single morphemes). Large structures such as embedded clauses, coordinated clauses, DPs, relative clauses etc. are sometimes more amenable to constituency tests (Extraction, superiority, C-command, principles A,B,C, polarity etc etc) and can be modified a lot more easily.
(4) If you choose almost anything, if you scratch deep enough, you will find interesting effects that haven't been explained yet. So perhaps stop worrying about WHAT to study and just start ANYWHERE.
(5) point 4 notwithstanding, starting anywhere can be a bit scary. So it is sometimes a good idea to find another research paper that you have studied and understand and enjoy and try and see if the analysis works the same way in "your" chosen language. E.g. you might find yourself asking things like: "I wonder if ellipsis works in my language the same way that Merchant says it does?" or "If Kayne is right about antisymmetry, then how would an antisymmetric analysis work in my language?" or "if extraposed DPs can escape islands in Spanish, I wonder if the same thing happens in my language?". Often, analyses aren't always an easy fit onto other languages and you will find that you will need to tweak the analysis a bit to get it to fit your language: and as you make those tweaks, you will start learning new things.
(5b) This is a variation of point 5. Look at older papers from the 80s or even the 70s. Often there is interesting data and analyses that need to be updated or brought into line with minimalist theory etc. So you could take an insight from an earlier paper and then try and retheorize it with more modern frameworks.
(6) If you are still wondering what constructions to look at, try looking through a syntax textbook for the canonical constructions that are usually looked at. (However, my feeling is that the examples in textbooks are often the ones that are highly studied and so it seems difficult to be able to find something novel to study about it.)
(7) Some authors are known for including interesting and strange data in their papers. For English, I've always found papers by Postal and Cullicover and also Richard Kayne to have wonderful little nuggets.
(8) Be specific. Try this step after you have narrowed down your search somewhat. Instead of asking " how do questions work in Bulgarian?" try asking "how do adjunct questions work in embedded clauses in bulgarian?'; Instead of asking "How does ellipsis work in Georgian?" consider asking "How does ellipsis work in the language of small children in Georgian?" (or whatever. I just made these up so don't place too much hope in them).
Of course, the common thread to all of these? READ!
Here is a list of potential relevant papers and/or books that are quite cited:
Dekkers, J., vd van der Leeuw, F., & vd van de Weijer, J. (2000). Optimality Theory: Phonology, Syntax and Acquisition (Oxford Linguistics).
Fawcett, R. P. (2000). A theory of syntax for systemic functional linguistics (Vol. 206). John Benjamins Publishing.
Gries, S. T., & Stefanowitsch, A. (Eds.). (2007). Corpora in cognitive linguistics: corpus-based approaches to syntax and lexis (Vol. 1). Walter de Gruyter.
Meurers, W. D. (2005). On the use of electronic corpora for theoretical linguistics: Case studies from the syntax of German. Lingua, 115(11), 1619-1639.
Friedemann, M. A., & Rizzi, L. (2014). The Acquisition of Syntax: studies in comparative developmental linguistics. Routledge.
Langendoen, D. T. (1969). The Study of Syntax: The Generative-Transformational Approach to the Structure of American English. Transatlantic Series in Linguistics.
Morley, G. D. (2000). Syntax in functional grammar: an introduction to lexicogrammar in systemic linguistics. A&C Black.
I hope that these references will serve as a good starting point.
This practical coursebook introduces all the basics of modern syntactic analysis in a simple step-by-step fashion. Each unit is constructed so that the reader discovers new ideas, formulates hypotheses and practises fundamentals. The reader is presented with short sections of explanation with examples, followed by practice exercises. Feedback and comment sections follow to enable students to monitor their progress. No previous background in syntax is assumed. Students move through all the key topics in the field including features, rules of combination and displacement, empty categories, and subcategorization. The theoretical perspective in this work is unique, drawing together the best ideas from three major syntactic frameworks (minimalism, HPSG and LFG). Students using this book will learn fundamentals in such a way that they can easily go on to pursue further study in any of these frameworks.
One of the courses that I take in my first year of Ph.D study is syntax, to be honest, it is really hard if you have not prepared with sufficient background knowledge. So my personal experience is that first you have to arm yourself with all the necessary syntactic weaponry, then you can proceed to more particular issues (especially about your native language, it is the easiest route to embark on the syntactic journey). So basically you have to read a lot of papers and books, then choose the one issue (or problem in other people's arguments) that most interests you. I hope this can of help to you, and good luck!
It is hard to find a research topic. Here are some practical ideas to get you started.
(1) Asking a supervisor or staff member for some suggestions to get you started is a good idea. Usually professors want you to succeed and will assist with some general ideas. BUT: Professors hate it when you walk into their offices without ideas or not appearing to be prepared. They want to know that you are actively thinking and that you are not trying to just be lazy. So I advise that you go through each of the steps below and generate 3 or 4 possible kinds of studies. THEN take these ideas to your professor or tutor and see what they suggest.
(2) Try and avoid English: if you speak any other language, then consider doing a research topic in that language. English is a wonderful language, but it is very intensively studied and all textbooks are full of it. So why join the herd? Draw on other interesting languages that you speak or which are spoken around you!
(3) As a rule of thumb, if you are a new linguist, try and choose a topic involving fairly large structures (i.e. not just single words or single morphemes). Large structures such as embedded clauses, coordinated clauses, DPs, relative clauses etc. are sometimes more amenable to constituency tests (Extraction, superiority, C-command, principles A,B,C, polarity etc etc) and can be modified a lot more easily.
(4) If you choose almost anything, if you scratch deep enough, you will find interesting effects that haven't been explained yet. So perhaps stop worrying about WHAT to study and just start ANYWHERE.
(5) point 4 notwithstanding, starting anywhere can be a bit scary. So it is sometimes a good idea to find another research paper that you have studied and understand and enjoy and try and see if the analysis works the same way in "your" chosen language. E.g. you might find yourself asking things like: "I wonder if ellipsis works in my language the same way that Merchant says it does?" or "If Kayne is right about antisymmetry, then how would an antisymmetric analysis work in my language?" or "if extraposed DPs can escape islands in Spanish, I wonder if the same thing happens in my language?". Often, analyses aren't always an easy fit onto other languages and you will find that you will need to tweak the analysis a bit to get it to fit your language: and as you make those tweaks, you will start learning new things.
(5b) This is a variation of point 5. Look at older papers from the 80s or even the 70s. Often there is interesting data and analyses that need to be updated or brought into line with minimalist theory etc. So you could take an insight from an earlier paper and then try and retheorize it with more modern frameworks.
(6) If you are still wondering what constructions to look at, try looking through a syntax textbook for the canonical constructions that are usually looked at. (However, my feeling is that the examples in textbooks are often the ones that are highly studied and so it seems difficult to be able to find something novel to study about it.)
(7) Some authors are known for including interesting and strange data in their papers. For English, I've always found papers by Postal and Cullicover and also Richard Kayne to have wonderful little nuggets.
(8) Be specific. Try this step after you have narrowed down your search somewhat. Instead of asking " how do questions work in Bulgarian?" try asking "how do adjunct questions work in embedded clauses in bulgarian?'; Instead of asking "How does ellipsis work in Georgian?" consider asking "How does ellipsis work in the language of small children in Georgian?" (or whatever. I just made these up so don't place too much hope in them).
Of course, the common thread to all of these? READ!